Package 'WebPower'

Description

This is a collection of tools for conducting both basic and advanced statistical power analysis including correlation, proportion, t-test, one-way ANOVA, two-way ANOVA, linear regression, logistic regression, Poisson regression, mediation analysis, longitudinal data analysis, structural equation modeling and multilevel modeling. It also serves as the engineer for conducting power analysis online at https://webpower.psychstat.org.

Details

This is a collection of tools for conducting both basic and advanced statistical power analysis including correlation, proportion, t-test, one-way ANOVA, two-way ANOVA, linear regression, logistic regression, Poisson regression, mediation analysis, longitudinal data analysis, structural equation modeling and multilevel modeling. It also serves as the engineer for conducting power analysis online at https://webpower.psychstat.org.

References

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Description

• ID. The identification number of the subjects.

• cluster. The cluster number.

• score. The score of the subject.

• group. The group number.

Usage

CRT2

Format

An object of class data.frame with 8 rows and 4 columns.

Examples

#  ID cluster score group
#  1 1 6 0
#  2 1 2 0
#  3 2 6 1
#  4 2 5 1
#  5 3 1 0
#  6 3 4 0
#  7 4 6 1
#  8 4 4 1

Description

• ID. The identification number of the subjects.

• cluster. The cluster number.

• score. The score of the subject.

• group. The group number.

Usage

CRT3

Format

An object of class data.frame with 30 rows and 4 columns.

Examples

#  id cluster score group
#  1 1 1.93 0
#  2 1 1.51 0
#  3 1 2.13 0
#  4 1 2.96 0
#  5 1 3.84 0
#  6 2 3.36 1
#  7 2 3.13 1
#  8 2 1.71 1
#  9 2 3.1 1
#  10 2 2.53 1
#  11 3 2.01 2
#  12 3 4.73 2
#  13 3 3.34 2
#  14 3 0.11 2
#  15 3 3.6 2
#  16 4 2 0
#  17 4 1.99 0
#  18 4 1.89 0
#  19 4 2.25 0
#  20 4 1.83 0
#  21 5 3.03 1
#  22 5 2.08 1
#  23 5 1.5 1
#  24 5 3.18 1
#  25 5 1.92 1
#  26 6 3.49 2
#  27 6 3.08 2
#  28 6 4.54 2
#  29 6 2.34 2
#  30 6 4.33 2

Description

Estimate multilevel effect size from data

Usage

estCRT2arm(file)
estCRT3arm(file)
estMRT2arm(file)
estMRT3arm(file)

Arguments

Description

• ID. The identification number of the subjects.

• cluster. The cluster number.

• score. The score of the subject.

• group. The group number.

Usage

MRT2

Format

An object of class data.frame with 16 rows and 4 columns.

Examples

#Example data for MRT with 2 arms
#  id cluster score group
#  1 1 6 0
#  2 1 2 0
#  3 1 3 1
#  4 1 3 1
#  5 2 6 0
#  6 2 10 0
#  7 2 7 1
#  8 2 6 1
#  9 3 6 0
#  10 3 5 0
#  11 3 4 1
#  12 3 4 1
#  13 4 1 0
#  14 4 8 0
#  15 4 10 1
#  16 4 -2 1

Description

• ID. The identification number of the subjects.

• cluster. The cluster number.

• score. The score of the subject.

• group. The group number.

Usage

MRT3

Format

An object of class data.frame with 24 rows and 4 columns.

Examples

#  id cluster score group
#  1 1 2 0
#  2 1 3 0
#  3 1 2 1
#  4 1 0 1
#  5 1 3 2
#  6 1 2 2
#  7 2 1 0
#  8 2 4 0
#  9 2 2 1
#  10 2 3 1
#  11 2 3 2
#  12 2 1 2
#  13 3 1 0
#  14 3 4 0
#  15 3 1 1
#  16 3 1 1
#  17 3 2 2
#  18 3 0 2
#  19 4 4 0
#  20 4 3 0
#  21 4 1 1
#  22 4 3 1
#  23 4 3 2
#  24 4 3 2

Description

The function searches in an interval for a root (i.e., zero) of the function f with respect to its first argument. The argument interval is for the input of x, the corresponding outcome interval will be used as the interval to be searched in.

Usage

nuniroot(f, interval, maxlength = 100)

Arguments

Value

A list with at least four components: root and f.root give the location of the root and the value of the function evaluated at that point. iter and estim.prec give the number of iterations used and an approximate estimated precision for root. (If the root occurs at one of the endpoints, the estimated precision is NA.)

Examples

f <- function(x) 1+x-0.5*x^2
interval <- c(-3,6)
nuniroot(f,interval)

Description

This function is used to plot the power analysis results for Latent Change Score Models.

Usage

## S3 method for class 'lcs.power'
plot(x, parameter, ...)

Arguments

References

Zhang, Z., & Liu, H. (2018). Sample Size and Measurement Occasion Planning for Latent Change Score Models through Monte Carlo Simulation. In E. Ferrer, S. M. Boker, and K. J. Grimm (Eds.) Advances in Longitudinal Models for Multivariate Psychology: A Festschrift for Jack McArdle.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Description

This function is used to plot the power curves generated by webpower.

Usage

## S3 method for class 'webpower'
plot(x, xvar = NULL, yvar = NULL, xlab = NULL,
  ylab = NULL, ...)

Arguments

Value

The plot.

Examples

res <- wp.correlation(n=seq(50,100,10),r=0.3, alternative="two.sided")
plot(res)

Description

This function is used to summary the power analysis results.

Usage

## S3 method for class 'webpower'
print(x, ...)

Arguments

Value

The printing of the input object of power analysis.

Examples

res <- wp.correlation(n=50,r=0.3, alternative="two.sided")
print(res)

Description

This function is for calculating SEM effect size.

Usage

sem.effect.size(full.model.pop, reduced.model)

Arguments

Value

An object of the power analysis.

References

Demidenko, E. (2007). Sample size determination for logistic regression revisited. Statistics in medicine, 26(18), 3385-3397.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

full.model.pop <-'
y1 ~ 0.4*x
y2 ~ 0.5*x + 0.2*y1
y3 ~ 0.4*x
y4 ~ 0.4*y1 + 0.4*y2 + 0.4*y3
y1 ~~ 0.84*y1
y2 ~~ 0.61*y2
y3 ~~ 0.84*y3
y4 ~~ 0.27*y4
'

reduced.model <-'
y1 ~ x
y2 ~ x 
y3 ~ x
y4 ~ y1 + y3
'

sem.effect.size(full.model.pop, reduced.model)

Description

This function is used to summary the power analysis results.

Usage

## S3 method for class 'power'
summary(object, ...)

Arguments

Value

The summary of the input object of power analysis.

Description

One-way analysis of variance (one-way ANOVA) is a technique used to compare means of two or more groups (e.g., Maxwell & Delaney, 2003). The ANOVA tests the null hypothesis that samples in two or more groups are drawn from populations with the same mean values. The ANOVA analysis typically produces an F-statistic, the ratio of the bewteen-group variance to the within-group variance.

Usage

wp.anova(k = NULL, n = NULL, f = NULL, alpha = 0.05, power = NULL,
  type = c("overall", "two.sided", "greater", "less"))

Arguments

Value

An object of the power analysis.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed). Hillsdale, NJ: Lawrence Erlbaum Associates.

Maxwell, S. E., & Delaney, H. D. (2004). Designing experiments and analyzing data: A model comparison perspective (Vol. 1). Psychology Press.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

#To calculate the statistical power for the overall test of one-way ANOVA:
wp.anova(f=0.25,k=4, n=100, alpha=0.05)
#  Power for One-way ANOVA
#
#    k   n    f alpha     power
#    4 100 0.25  0.05 0.5181755
#
#  NOTE: n is the total sample size (overall)
#  URL: http://psychstat.org/anova

#To calculate the power curve with a sequence of sample sizes:
res <- wp.anova(f=0.25, k=4, n=seq(100,200,10), alpha=0.05)
res
#  Power for One-way ANOVA
#
#    k   n    f alpha     power
#    4 100 0.25  0.05 0.5181755
#    4 110 0.25  0.05 0.5636701
#    4 120 0.25  0.05 0.6065228
#    4 130 0.25  0.05 0.6465721
#    4 140 0.25  0.05 0.6837365
#    4 150 0.25  0.05 0.7180010
#    4 160 0.25  0.05 0.7494045
#    4 170 0.25  0.05 0.7780286
#    4 180 0.25  0.05 0.8039869
#    4 190 0.25  0.05 0.8274169
#    4 200 0.25  0.05 0.8484718
#
#  NOTE: n is the total sample size (overall)
#  URL: http://psychstat.org/anova

#To plot the power curve:
plot(res, type='b')

#To estimate the sample size with a given power:
wp.anova(f=0.25,k=4, n=NULL, alpha=0.05, power=0.8)
#  Power for One-way ANOVA
#
#    k        n    f alpha power
#    4 178.3971 0.25  0.05   0.8
#
#  NOTE: n is the total sample size (overall)
#  URL: http://psychstat.org/anova

#To estimate the minimum detectable effect size with a given power:
wp.anova(f=NULL,k=4, n=100, alpha=0.05, power=0.8)
#  Power for One-way ANOVA
#
#    k   n         f alpha power
#    4 100 0.3369881  0.05   0.8
#
#  NOTE: n is the total sample size (overall)
#  URL: http://psychstat.org/anova

#To conduct power analysis for a contrast one-way ANOVA:
wp.anova(f=0.25,k=4, n=100, alpha=0.05, type='two.sided')
#  Power for One-way ANOVA
#
#    k   n    f alpha     power
#    4 100 0.25  0.05 0.6967142
#
#  NOTE: n is the total sample size (contrast, two.sided)
#  URL: http://psychstat.org/anova

#To calculate the power curve with a sequence of sample sizes:
res <- wp.anova(f=seq(0.1, 0.8, 0.1), k=4, n=100, alpha=0.05)
res
#  Power for One-way ANOVA
#
#    k   n   f alpha     power
#    4 100 0.1  0.05 0.1128198
#    4 100 0.2  0.05 0.3452612
#    4 100 0.3  0.05 0.6915962
#    4 100 0.4  0.05 0.9235525
#    4 100 0.5  0.05 0.9911867
#    4 100 0.6  0.05 0.9995595
#    4 100 0.7  0.05 0.9999908
#    4 100 0.8  0.05 0.9999999
#
#  NOTE: n is the total sample size (overall)
#  URL: http://psychstat.org/anova

Description

The power analysis procedure for one-way ANOVA with binary data is introduced by Mai and Zhang (2017). One-way ANOVA with binary data is used for comparing means of three or more groups of binary data. Its outcome variable is supposed to follow Bernoulli distribution. And its overall test uses a likelihood ratio test statistics.

Usage

wp.anova.binary(k = NULL, n = NULL, V = NULL, alpha = 0.05,
  power = NULL)

Arguments

Value

An object of the power analysis.

References

Mai, Y., & Zhang, Z. (2017). Statistical Power Analysis for Comparing Means with Binary or Count Data Based on Analogous ANOVA. In L. A. van der Ark, M. Wiberg, S. A. Culpepper, J. A. Douglas, & W.-C. Wang (Eds.), Quantitative Psychology - The 81st Annual Meeting of the Psychometric Society, Asheville, North Carolina, 2016: Springer.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

#To calculate the statistical power for one-way ANOVA (overall test) with binary data:
wp.anova.binary(k=4,n=100,V=0.15,alpha=0.05)
#  One-way Analogous ANOVA with Binary Data
#
#    k   n    V alpha     power
#    4 100 0.15  0.05 0.5723443
#
#  NOTE: n is the total sample size
#  URL: http://psychstat.org/anovabinary

#To generate a power curve given a sequence of sample sizes:
res <- wp.anova.binary(k=4,n=seq(100,200,10),V=0.15,alpha=0.05,power=NULL)
res
#  One-way Analogous ANOVA with Binary Data
#
#    k   n    V alpha     power
#    4 100 0.15  0.05 0.5723443
#    4 110 0.15  0.05 0.6179014
#    4 120 0.15  0.05 0.6601594
#    4 130 0.15  0.05 0.6990429
#    4 140 0.15  0.05 0.7345606
#    4 150 0.15  0.05 0.7667880
#    4 160 0.15  0.05 0.7958511
#    4 170 0.15  0.05 0.8219126
#    4 180 0.15  0.05 0.8451603
#    4 190 0.15  0.05 0.8657970
#    4 200 0.15  0.05 0.8840327
#
#  NOTE: n is the total sample size
#  URL: http://psychstat.org/anovabinary

#To plot the power curve:
plot(res) 

#To calculate the required sample size for one-way ANOVA (overall test) with binary data:
wp.anova.binary(k=4,n=NULL,V=0.15,power=0.8, alpha=0.05)
#  One-way Analogous ANOVA with Binary Data
#
#    k        n    V alpha power
#    4 161.5195 0.15  0.05   0.8
#
#  NOTE: n is the total sample size
#  URL: http://psychstat.org/anovabinary

#To calculate the minimum detectable effect size for one-way ANOVA (overall test) with binary data:
wp.anova.binary(k=4,n=100,V=NULL,power=0.8, alpha=0.05)
#  One-way Analogous ANOVA with Binary Data
#
#    k   n         V alpha power
#    4 100 0.1906373  0.05   0.8
#
#  NOTE: n is the total sample size
#  URL: http://psychstat.org/anovabinary

#To generate a power curve given a sequence of effect sizes:
wp.anova.binary(k=4,n=100,V=seq(0.1,0.5,0.05),alpha=0.05,power=NULL)
#  One-way Analogous ANOVA with Binary Data
#
#    k   n    V alpha     power
#    4 100 0.10  0.05 0.2746396
#    4 100 0.15  0.05 0.5723443
#    4 100 0.20  0.05 0.8402271
#    4 100 0.25  0.05 0.9659434
#    4 100 0.30  0.05 0.9961203
#    4 100 0.35  0.05 0.9997729
#    4 100 0.40  0.05 0.9999933
#    4 100 0.45  0.05 0.9999999
#    4 100 0.50  0.05 1.0000000
#
#  NOTE: n is the total sample size
#  URL: http://psychstat.org/anovabinary

Description

The power analysis procedure for one-way ANOVA with count data is introduced by Mai and Zhang (2017). One-way ANOVA with count data is used for comparing means of three or more groups of binary data. Its outcome variable is supposed to follow Poisson distribution. And its overall test uses a likelihood ratio test statistics.

Usage

wp.anova.count(k = NULL, n = NULL, V = NULL, alpha = 0.05,
  power = NULL)

Arguments

Value

An object of the power analysis.

References

Mai, Y., & Zhang, Z. (2017). Statistical Power Analysis for Comparing Means with Binary or Count Data Based on Analogous ANOVA. In L. A. van der Ark, M. Wiberg, S. A. Culpepper, J. A. Douglas, & W.-C. Wang (Eds.), Quantitative Psychology - The 81st Annual Meeting of the Psychometric Society, Asheville, North Carolina, 2016: Springer.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

#To calculate the statistical power for one-way ANOVA (overall test) with count data:
wp.anova.count(k=4,n=100,V=0.148,alpha=0.05)
#  One-way Analogous ANOVA with Count Data
#
#    k   n     V alpha     power
#    4 100 0.148  0.05 0.5597441
#
#  NOTE: n is the total sample size
#  URL: http://psychstat.org/anovacount

#To generate a power curve given sequence of sample sizes:
res <- wp.anova.count(k=4,n=seq(100,200,10),V=0.148,alpha=0.05,power=NULL)
res
#  One-way Analogous ANOVA with Count Data
#
#		k   n     V alpha     power
#		4 100 0.148  0.05 0.5597441
#		4 110 0.148  0.05 0.6049618
#		4 120 0.148  0.05 0.6470911
#		4 130 0.148  0.05 0.6860351
#		4 140 0.148  0.05 0.7217782
#		4 150 0.148  0.05 0.7543699
#		4 160 0.148  0.05 0.7839101
#		4 170 0.148  0.05 0.8105368
#		4 180 0.148  0.05 0.8344142
#		4 190 0.148  0.05 0.8557241
#		4 200 0.148  0.05 0.8746580
#
#  NOTE: n is the total sample size
#  URL: http://psychstat.org/anovacount

#To plot the power curve:
plot(res)

#To calculate the required sample size for one-way ANOVA (overall test) with count data:
wp.anova.count(k=4,n=NULL,V=0.148,power=0.8, alpha=0.05)
#  One-way Analogous ANOVA with Count Data
#
#    k        n     V alpha power
#    4 165.9143 0.148  0.05   0.8
#
#  NOTE: n is the total sample size
#  URL: http://psychstat.org/anovacount

#To calculate the minimum detectable effect size for one-way ANOVA (overall test) with count data:
wp.anova.count(k=4,n=100,V=NULL,power=0.8, alpha=0.05)
#  One-way Analogous ANOVA with Count Data
#
#    k   n         V alpha power
#    4 100 0.1906373  0.05   0.8
#
#  NOTE: n is the total sample size
#  URL: http://psychstat.org/anovacount

#To generate a power curve given a sequence of effect sizes:
res <- wp.anova.count(k=5,n=100,V=seq(0.1,0.5,0.05),alpha=0.05,power=NULL)
res
#  One-way Analogous ANOVA with Count Data
#
#    k   n    V alpha     power
#    5 100 0.10  0.05 0.3200744
#    5 100 0.15  0.05 0.6634861
#    5 100 0.20  0.05 0.9118531
#    5 100 0.25  0.05 0.9893643
#    5 100 0.30  0.05 0.9994549
#    5 100 0.35  0.05 0.9999887
#    5 100 0.40  0.05 0.9999999
#    5 100 0.45  0.05 1.0000000
#    5 100 0.50  0.05 1.0000000
#
#  NOTE: n is the total sample size
#  URL: http://psychstat.org/anovacount

Description

A longitudinal design often involves data collection on multiple variables from multiple participants at multiple times. Growth curve models (GCM) are structural equation models for longitudinal data analysis (McArdle & Epstein, 1987; McArdle & Nesselroade, 2014). Latent change score models (LCSM) combine difference equations with growth curves to investigate change in longitudinal studies . LCSM provied an efficient way to model nonlinear trajectory (e.g., McArdle, 2000; McArdle & Hamagami, 2001; Hamagami et al., 2010). This function is used to conduct power analysis for bivariate LCSMs based on a Monte Carlo method ( a method also used by Muthén & Muthén, 2002; Thoemmes et al., 2010; Zhang & Wang, 2009; Zhang, 2014). For each Monte Carlo replication, the Maximum likelihood ratio test is used for the model, while the Wald test is used for the parameter test. The method can obtain the power for testing each individual parameter of the models such as the change rate and coupling parameters.

Usage

wp.blcsm(N = 100, T = 5, R = 1000, betay = 0, my0 = 0, mys = 0,
  varey = 1, vary0 = 1, varys = 1, vary0ys = 0, alpha = 0.05,
  betax = 0, mx0 = 0, mxs = 0, varex = 1, varx0 = 1, varxs = 1,
  varx0xs = 0, varx0y0 = 0, varx0ys = 0, vary0xs = 0, varxsys = 0,
  gammax = 0, gammay = 0, ...)

Arguments

Value

An object of the power analysis. The output of the R function includes 4 main pieces of information for each parameter in the model. The first is the Monte Carlo estimate (mc.est). It is calculated as the mean of the R sets of parameter estimates from the simulated data. Note that the Monte Carlo estimates should be close to the population parameter values used in the model. The second is the Monte Carlo standard deviation (mc.sd), which is calculated as the standard deviation of the R sets of parameter estimates. The third is the Monte Carlo standard error (mc.se), which is obtained as the average of the R sets of standard error estimates of the parameter estimates. Lastly, mc.power is the statistical power for each parameter.

References

Zhang, Z., & Liu, H. (2018). Sample Size and Measurement Occasion Planning for Latent Change Score Models through Monte Carlo Simulation. In E. Ferrer, S. M. Boker, and K. J. Grimm (Eds.) Advances in Longitudinal Models for Multivariate Psychology: A Festschrift for Jack McArdle.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

## Not run: 
#To conduct power analysis for a bivariate LCSM with sample size equal to 100:
wp.blcsm(N=100, T=5, R=1000, betay=0.08, my0=20, mys=1.5, varey=9,
     vary0=3, varys=1, vary0ys=0, alpha=0.05, betax=0.2, mx0=20, mxs=5,
         varex=9, varx0=3, varxs=1, varx0xs=0, varx0y0=1, varx0ys=0,
                            vary0xs=0, varxsys=0, gammax=0, gammay=-.1)
#             pop.par mc.est  mc.sd  mc.se  mc.power N T
#    betax    0.20     0.230  0.260  0.187  0.241   100 5
#    betay    0.08     0.164  0.572  0.435  0.081   100 5
#    gammax   0.00    -0.033  0.234  0.178  0.112   100 5
#    gammay  -0.10    -0.175  0.641  0.458  0.075   100 5
#    mx0     20.00    20.004  0.336  0.326  1.000   100 5
#    mxs      5.00     5.933  7.848  5.615  0.167   100 5
#    my0     20.00    20.019  0.346  0.326  1.000   100 5
#    mys      1.50     0.451  6.933  5.321  0.156   100 5
#    varex    9.00     8.941  0.744  0.732  1.000   100 5
#    varey    9.00     8.939  0.749  0.720  1.000   100 5
#    varx0    3.00     3.029  1.243  1.222  0.739   100 5
#    varx0xs  0.00    -0.210  0.768  0.767  0.030   100 5
#    varx0y0  1.00     1.052  0.840  0.835  0.226   100 5
#    varx0ys  0.00    -0.012  0.668  0.601  0.017   100 5
#    varxs    0.60     2.343  6.805  2.687  0.090   100 5
#    varxsys  0.00     0.072  3.559  1.740  0.019   100 5
#    vary0    3.00     2.951  1.423  1.245  0.684   100 5
#    vary0xs  0.00     0.198  2.263  1.629  0.031   100 5
#    vary0ys  0.00    -0.371  1.970  1.511  0.106   100 5
#    varys    0.05     1.415  3.730  2.096  0.024   100 5

#To conduct power analysis for a bivariate LCSM with sample size equal to 500:
wp.blcsm(N=500, T=5, R=1000, betay=0.08, my0=20, mys=1.5, varey=9,
      vary0=3, varys=1, vary0ys=0, alpha=0.05, betax=0.2, mx0=20
           , mxs=5, varex=9, varx0=3, varxs=1, varx0xs=0, varx0y0=1,
                 varx0ys=0, vary0xs=0, varxsys=0, gammax=0, gammay=-.1)
#           pop.par mc.est mc.sd mc.se  mc.power N  T
#    betax    0.20  0.2009 0.031 0.031   1.000  500 5
#    betay    0.08  0.0830 0.070 0.068   0.199  500 5
#    gammax   0.00 -0.0014 0.030 0.029   0.057  500 5
#    gammay  -0.10 -0.1022 0.072 0.073   0.271  500 5
#    mx0     20.00 19.9911 0.145 0.145   1.000  500 5
#    mxs      5.00  5.0308 0.939 0.942   1.000  500 5
#    my0     20.00 19.9999 0.143 0.146   1.000  500 5
#    mys      1.50  1.4684 0.889 0.885   0.420  500 5
#    varex    9.00  8.9836 0.340 0.328   1.000  500 5
#    varey    9.00  8.9961 0.341 0.328   1.000  500 5
#    varx0    3.00  3.0052 0.524 0.523   1.000  500 5
#    varx0xs  0.00 -0.0144 0.222 0.230   0.047  500 5
#    varx0y0  1.00  1.0064 0.360 0.360   0.808  500 5
#    varx0ys  0.00 -0.0012 0.199 0.201   0.051  500 5
#    varxs    1.00  1.0312 0.180 0.189   1.000  500 5
#    varxsys  0.00  0.0028 0.161 0.163   0.045  500 5
#    vary0    3.00  2.9777 0.519 0.547   1.000  500 5
#    vary0xs  0.00  0.0072 0.286 0.294   0.035  500 5
#    vary0ys  0.00 -0.0135 0.252 0.257   0.043  500 5
#    varys    1.00  1.0246 0.260 0.253   0.999  500 5

## End(Not run)

Description

This function is for power analysis for correlation. Correlation measures whether and how a pair of variables are related. The Pearson Product Moment correlation coefficient (r) is adopted here. The power calculation for correlation is conducted based on Fisher's z transformation of Pearson correlation coefficent (Fisher, 1915, 1921).

Usage

wp.correlation(n = NULL, r = NULL, power = NULL, p = 0, rho0 = 0,
  alpha = 0.05, alternative = c("two.sided", "less", "greater"))

Arguments

Value

An object of the power analysis.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed). Hillsdale, NJ: Lawrence Erlbaum Associates.

Fisher, R. A. (1915). Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika, 10(4), 507-521.

Fisher, R. A. (1921). On the probable error of a coefficient of correlation deduced from a small sample. Metron, 1, 3-32.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

wp.correlation(n=50,r=0.3, alternative="two.sided")
#  Power for correlation
#
#     n   r alpha     power
#    50 0.3  0.05 0.5728731
#        
#  URL: http://psychstat.org/correlation

#To calculate the power curve with a sequence of sample sizes:
res <- wp.correlation(n=seq(50,100,10),r=0.3, alternative="two.sided")
res
#  Power for correlation
#
#      n   r alpha     power
#     50 0.3  0.05 0.5728731
#     60 0.3  0.05 0.6541956
#     70 0.3  0.05 0.7230482
#     80 0.3  0.05 0.7803111
#     90 0.3  0.05 0.8272250
#    100 0.3  0.05 0.8651692
#
#  URL: http://psychstat.org/correlation

#To plot the power curve:
plot(res, type='b')

#To estimate the sample size with a given power:
wp.correlation(n=NULL, r=0.3, power=0.8, alternative="two.sided")
#  Power for correlation
#
#           n   r alpha power
#    83.94932 0.3  0.05   0.8
#
#  URL: http://psychstat.org/correlation

#To estimate the minimum detectable effect size with a given power:
wp.correlation(n=NULL,r=0.3, power=0.8, alternative="two.sided")
#  Power for correlation
#
#           n   r alpha power
#    83.94932 0.3  0.05   0.8
#
#  URL: http://psychstat.org/correlation
#
#To calculate the power curve with a sequence of effect sizes:
res <- wp.correlation(n=100,r=seq(0.05,0.8,0.05), alternative="two.sided")
res
#	 Power for correlation
#
#      n    r alpha      power
#    100 0.05  0.05 0.07854715
#    100 0.10  0.05 0.16839833
#    100 0.15  0.05 0.32163978
#    100 0.20  0.05 0.51870091
#    100 0.25  0.05 0.71507374
#    100 0.30  0.05 0.86516918
#    100 0.35  0.05 0.95128316
#    100 0.40  0.05 0.98724538
#    100 0.45  0.05 0.99772995
#    100 0.50  0.05 0.99974699
#    100 0.55  0.05 0.99998418
#    100 0.60  0.05 0.99999952
#    100 0.65  0.05 0.99999999
#    100 0.70  0.05 1.00000000
#    100 0.75  0.05 1.00000000
#    100 0.80  0.05 1.00000000
#
#	 URL: http://psychstat.org/correlation

Description

Cluster randomized trials (CRT) are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a contral arm (Liu, 2013). The data from CRT can be analyzed in a two-level hierachical linear model, where the indicator variable for treatment assignment is included in second level. If a study contains multiple treatments, then mutiple indicators will be used. This function is for designs with 2 arms (i.e., a treatment and a control). Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013).

Usage

wp.crt2arm(n = NULL, f = NULL, J = NULL, icc = NULL, power = NULL,
  alpha = 0.05, alternative = c("two.sided", "one.sided"), interval = NULL)

Arguments

Value

An object of the power analysis.

References

Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.

Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

#To calculate the statistical power given sample size and effect size:
wp.crt2arm(f = 0.6, n = 20, J = 10, icc = 0.1, alpha = 0.05, power = NULL)
#  Cluster randomized trials with 2 arms
#
#     J  n   f icc     power alpha
#    10 20 0.6 0.1 0.5901684  0.05
#
#  NOTE: n is the number of subjects per cluster.
#  URL: http://psychstat.org/crt2arm

#To generate a power curve given a sequence of sample sizes:
res <- wp.crt2arm(f = 0.6, n = seq(20,100,10), J = 10,
                       icc = 0.1, alpha = 0.05, power = NULL)
res
#  Cluster randomized trials with 2 arms
#
#     J   n   f icc     power alpha
#    10  20 0.6 0.1 0.5901684  0.05
#    10  30 0.6 0.1 0.6365313  0.05
#    10  40 0.6 0.1 0.6620030  0.05
#    10  50 0.6 0.1 0.6780525  0.05
#    10  60 0.6 0.1 0.6890755  0.05
#    10  70 0.6 0.1 0.6971076  0.05
#    10  80 0.6 0.1 0.7032181  0.05
#    10  90 0.6 0.1 0.7080217  0.05
#    10 100 0.6 0.1 0.7118967  0.05
#
#  NOTE: n is the number of subjects per cluster.
#  URL: http://psychstat.org/crt2arm

#To plot the power curve:
plot(res) 

#To calculate the required sample size given power and effect size:
wp.crt2arm(f = 0.8, n = NULL, J = 10,
                 icc = 0.1, alpha = 0.05, power = 0.8)
#  Cluster randomized trials with 2 arms
#
#     J        n   f icc power alpha
#    10 16.02558 0.8 0.1   0.8  0.05
#
#  NOTE: n is the number of subjects per cluster.
#  URL: http://psychstat.org/crt2arm

Description

Cluster randomized trials (CRT) are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a control arm (Liu, 2013). The data from CRT can be analyzed in a two-level hierarchical linear model, where the indicator variable for treatment assignment is included in second level. If a study contains multiple treatments, then multiple indicators will be used. This function is for designs with 3 arms (i.e., two treatments and a control). Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013).

Usage

wp.crt3arm(n = NULL, f = NULL, J = NULL, icc = NULL, power = NULL,
  alpha = 0.05, alternative = c("two.sided", "one.sided"),
  type = c("main", "treatment", "omnibus"), interval = NULL)

Arguments

Value

An object of the power analysis.

References

Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.

Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

#To calculate the statistical power given sample size and effect size:
wp.crt3arm(f = 0.5, n = 20, J = 10, icc = 0.1, alpha = 0.05, power = NULL)
#  Cluster randomized trials with 3 arms
#
#     J  n   f icc     power alpha
#    10 20 0.5 0.1 0.3940027  0.05
#
#  NOTE: n is the number of subjects per cluster.
#  URL: http://psychstat.org/crt3arm

#To generate a power curve given a sequence of sample sizes:
res <- wp.crt3arm(f = 0.5, n = seq(20, 100, 10), J = 10,
                       icc = 0.1, alpha = 0.05, power = NULL)
res
#  Cluster randomized trials with 3 arms
#
#     J   n   f icc     power alpha
#    10  20 0.5 0.1 0.3940027  0.05
#    10  30 0.5 0.1 0.4304055  0.05
#    10  40 0.5 0.1 0.4513376  0.05
#    10  50 0.5 0.1 0.4649131  0.05
#    10  60 0.5 0.1 0.4744248  0.05
#    10  70 0.5 0.1 0.4814577  0.05
#    10  80 0.5 0.1 0.4868682  0.05
#    10  90 0.5 0.1 0.4911592  0.05
#    10 100 0.5 0.1 0.4946454  0.05
#
#  NOTE: n is the number of subjects per cluster.
#  URL: http://psychstat.org/crt3arm

#To plot the power curve:
plot(res)

#To calculate the required sample size given power and effect size:
wp.crt3arm(f = 0.8, n = NULL, J = 10, icc = 0.1, alpha = 0.05, power = 0.8)
#  Cluster randomized trials with 3 arms
#
#     J        n   f icc power alpha
#    10 27.25145 0.8 0.1   0.8  0.05
#
#  NOTE: n is the number of subjects per cluster.
#  URL: http://psychstat.org/crt3arm

Description

This function is for effect size and ICC calculation for CRT with 2 arms based on empirical data. Cluster randomized trials (CRT) are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a contral arm (Liu, 2013). The data from CRT can be analyzed in a two-level hierachical linear model, where the indicator variable for treatment assignment is included in second level. If a study contains multiple treatments, then mutiple indicators will be used. This function is for designs with 3 arms (i.e., two treatments and a control). Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013). The Effect size f specifies the main effect of treatment, the mean difference between the treatment clusters and the control clusters. This function is used to calculate the effect size with a input data set.

Usage

wp.effect.CRT2arm(file)

Arguments

Value

A list including effect size f and ICC.

References

Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.

Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.

Examples

#Empirical data set CRT2:
CRT2
#ID cluster score group
#1 1 6 0
#2 1 2 0
#3 2 6 1
#4 2 5 1
#5 3 1 0
#6 3 4 0
#7 4 6 1
#8 4 4 1

#To calculate the effect size and ICC based on empirical data
wp.effect.CRT2arm (CRT2)
#  Effect size for CRT2arm
#  
#         f  ICC
#  1.264911 -0.5
#  
#  NOTE: f is the effect size.
#  URL: http://psychstat.org/crt2arm

Description

This function is for effect size and ICC calculation for Cluster randomized trials (CRT) with 3 arms based on empirical data. CRT are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a contral arm (Liu, 2013). The data from CRT can be analyzed in a two-level hierachical linear model, where the indicator variable for treatment assignment is included in second level. If a study contains multiple treatments, then mutiple indicators will be used. This function is for designs with 3 arms (i.e., two treatments and a control). Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013). The Effect size f specifies the main effect of treatment, the mean difference between the treatment clusters and the control clusters. This function is used to calculate the effect size with a input data set.

Usage

wp.effect.CRT3arm(file)

Arguments

Value

A list including effect size f1, f2, f3, and ICC.

References

Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.

Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.

Examples

#To calculate the effect sizes based on empirical data
wp.effect.CRT3arm (CRT3)
#  Effect size for CRT3arm
#  
#  f1         f2        f3       ICC
#  0.6389258 -0.6189113 0.3931397 -0.019794
#  
#  NOTE: f1 for treatment main effect;
#        f2 for difference between two treatments;
#        f3 for effect size of omnibus test.
#  URL: http://psychstat.org/crt3arm

Description

This function is for effect size calculation for Multisite randomized trials (MRT) with 2 arms based on empirical data. MRTs are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a contral arm (Liu, 2013). The data from MRT can be analyzed in a two-level hierachical linear model, where the indicator variable for treatment assignment is included in first level. If a study contains multiple treatments, then mutiple indicators will be used. Three types of tests are considered in the function: (1) The "main" type tests treatment main effect; (2) The "site" type tests the variance of cluster/site means; and (3) The "variance" type tests variance of treatment effects. Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013). This function is used to calculate the effect size with a input data set.

Usage

wp.effect.MRT2arm(file)

Arguments

Value

A list including effect size f.

References

Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.

Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.

Examples

#To calculate the effect size based on empirical data
wp.effect.MRT2arm (MRT2)
#  Effect size for MRT2arm
#  
#  f
#  -0.2986755
#  
#  NOTE: f is the effect size.
#  URL: http://psychstat.org/mrt2arm

Description

This function is for effect size calculation for Multisite randomized trials (MRT) with 3 arms based on empirical data. MRTs are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a contral arm (Liu, 2013). The data from MRT can be analyzed in a two-level hierachical linear model, where the indicator variable for reatment assignment is included in first level. If a study contains multiple treatments, then mutiple indicators will be used. This function is for designs with 3 arms (i.e., two treatments and a control). Three types of tests are considered in the function: (1) The "main" type tests treatment main effect; (2) The "treatment" type tests the difference between the two treaments; and (3) The "omnibus" type tests whether the three arms are all equivalent. Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013). This function is used to calculate the effect size with a input data set.

Usage

wp.effect.MRT3arm(file)

Arguments

Value

A list including effect size f.

References

Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.

Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.

Examples

#To calculate the effect size and ICC based on empirical data
#  wp.effect.MRT3arm(MRT3)
#  Effect size for MRT3arm
#  
#  f1        f2
#  -0.6214215 -0.355098
#  
#  NOTE: f1 for treatment main effect;
#        f2 for comparing the two treatments.
#  URL: http://psychstat.org/mrt3arm

Description

Power analysis for two-way, three-way and k-way ANOVA

Usage

wp.kanova(n = NULL, ndf = NULL, f = NULL, ng = NULL, alpha = 0.05, power = NULL)

Arguments

References

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

########## Chapter 6. two-way and three-way ANOVA  #############
## Main effect of two-way ANOVA
wp.kanova(n=120, ndf=2, f=0.2, alph=0.05, ng=6)

## Interaction effect of two-way ANOVA
wp.kanova(n=120, ndf=2, f=0.4, alph=0.05, ng=6)

## Interaction effect of three-way ANOVA
wp.kanova(n=360, ndf=4, f=0.3, alph=0.05, ng=18)

Description

A longitudinal design often involves data collection on multiple variables from multiple participants at multiple times. Growth curve models (GCM) are structural equation models for longitudinal data analysis (McArdle & Epstein, 1987; McArdle & Nesselroade, 2014). Latent change score models (LCSM) combine difference equations with growth curves to investigate change in longitudinal studies . LCSM provied an efficient way to model nonlinear trajectory (e.g., McArdle, 2000; McArdle & Hamagami, 2001; Hamagami et al., 2010). This function is used to conduct power analysis for univariate LCSMs based on a Monte Carlo method ( a method also used by Muthén & Muthén, 2002; Thoemmes et al., 2010; Zhang & Wang, 2009; Zhang, 2014). For each Monte Carlo replication, the Maximum likelihood ratio test is used for the model, while the Wald test is used for the parameter test. The method can obtain the power for testing each individual parameter of the models such as the change rate and coupling parameters.

Usage

wp.lcsm(N = 100, T = 5, R = 1000, betay = 0, my0 = 0, mys = 0,
  varey = 1, vary0 = 1, varys = 1, vary0ys = 0, alpha = 0.05, ...)

Arguments

Value

An object of the power analysis. The output of the R function includes 4 main pieces of information for each parameter in the model. The first is the Monte Carlo estimate (mc.est). It is calculated as the mean of the R sets of parameter estimates from the simulated data. Note that the Monte Carlo estimates should be close to the population parameter values used in the model. The second is the Monte Carlo standard deviation (mc.sd), which is calculated as the standard deviation of the R sets of parameter estimates. The third is the Monte Carlo standard error (mc.se), which is obtained as the average of the R sets of standard error estimates of the parameter estimates. Lastly, mc.power is the statistical power for each parameter.

References

Zhang, Z., & Liu, H. (2018). Sample Size and Measurement Occasion Planning for Latent Change Score Models through Monte Carlo Simulation. In E. Ferrer, S. M. Boker, and K. J. Grimm (Eds.) Advances in Longitudinal Models for Multivariate Psychology: A Festschrift for Jack McArdle.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

## Not run: 
#Power analysis for a univariate LCSM
#Power for each parameter given sample size, number of measurement occasions,
# true effect (true values of parameters), and significance level:
wp.lcsm(N = 100, T = 5, R = 1000, betay = 0.1, my0 = 20, mys = 1.5,
        varey = 9, vary0 = 2.5, varys = .05, vary0ys = 0, alpha = 0.05)   
#           pop.par mc.est mc.sd mc.se mc.power  N  T
#    betay   0.10   0.103  0.043 0.044  0.664   100 5
#    my0    20.00  19.999  0.324 0.319  1.000   100 5
#    mys     1.50   1.418  1.106 1.120  0.274   100 5
#    varey   9.00   8.961  0.724 0.732  1.000   100 5
#    vary0   2.50   2.463  1.151 1.139  0.583   100 5
#    vary0ys 0.00  -0.004  0.408 0.403  0.048   100 5
#    varys   0.05   0.053  0.173 0.175  0.050   100 5
#  
#  #To calculate the Type I error rate and power for parameters
  wp.lcsm(N = 100, T = 5, R = 1000, betay = 0, my0 = 0, mys = 0,
              varey = 1, vary0 = 1, varys = 1, vary0ys = 0,alpha = 0.05)
#           pop.par	mc.est	mc.sd	mc.se	mc.power N	T
#    betay   0		0.001	0.056	0.056	0.046	100 5
#    my0		0		0.001	0.129	0.126	0.056	100 5
#    mys		0		0.002	0.105	0.105	0.044	100 5
#    varey   1		0.994	0.083	0.081	1.000	100 5
#    vary0   1		0.990	0.236	0.230	1.000	100 5
#    vary0ys	0		-0.005	0.136	0.136	0.044	100 5
#    varys   1		1.006	0.227	0.227	1.000	100 5

# To generate a power curve for different sample sizes for a univariate LCSM
res <- wp.lcsm(N = seq(100, 200, 10), T = 5, R = 1000, betay = 0.1,
                          my0 = 20, mys = 1.5, varey = 9, vary0 = 2.5,
                               varys = .05, vary0ys = 0, alpha = 0.05)
#plot(res, parameter='betay')
#plot(res, parameter='mys')

# To generate a power curve for different numbers of occasions for a univariate LCSM
res <- wp.lcsm(N = 100, T = 4:10, R = 1000, betay = 0.1, my0 = 20, mys = 1.5,
                varey = 9, vary0 = 2.5, varys = .05, vary0ys = 0, alpha = 0.05)
#plot(res, parameter='betay')
#plot(res, parameter='mys')

## End(Not run)

Description

This function is for Logistic regression models. Logistic regression is a type of generalized linear models where the outcome variable follows Bernoulli distribution. Here, Maximum likelihood methods is used to estimate the model parameters. The estimated regression coefficent is assumed to follow a normal distribution. A Wald test is use to test the mean difference between the estimated parameter and the null parameter (tipically the null hypothesis assumes it equals 0). The procedure introduced by Demidenko (2007) is adopted here for computing the statistical power.

Usage

wp.logistic(n = NULL, p0 = NULL, p1 = NULL, alpha = 0.05,
  power = NULL, alternative = c("two.sided", "less", "greater"),
  family = c("Bernoulli", "exponential", "lognormal", "normal", "Poisson",
  "uniform"), parameter = NULL)

Arguments

Value

An object of the power analysis.

References

Demidenko, E. (2007). Sample size determination for logistic regression revisited. Statistics in medicine, 26(18), 3385-3397.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

#To calculate the statistical power given sample size and effect size:
wp.logistic(n = 200, p0 = 0.15, p1 = 0.1, alpha = 0.05,
            power = NULL, family = "normal", parameter = c(0,1))
#  Power for logistic regression
#
#      p0  p1     beta0      beta1   n alpha     power
#    0.15 0.1 -1.734601 -0.4626235 200  0.05 0.6299315
#
#  URL: http://psychstat.org/logistic

#To generate a power curve given a sequence of sample sizes:
res <- wp.logistic(n = seq(100,500,50), p0 = 0.15, p1 = 0.1, alpha = 0.05,
                power = NULL, family = "normal", parameter = c(0,1))
res
#  Power for logistic regression
#
#      p0  p1     beta0      beta1   n alpha     power
#    0.15 0.1 -1.734601 -0.4626235 100  0.05 0.3672683
#    0.15 0.1 -1.734601 -0.4626235 150  0.05 0.5098635
#    0.15 0.1 -1.734601 -0.4626235 200  0.05 0.6299315
#    0.15 0.1 -1.734601 -0.4626235 250  0.05 0.7264597
#    0.15 0.1 -1.734601 -0.4626235 300  0.05 0.8014116
#    0.15 0.1 -1.734601 -0.4626235 350  0.05 0.8580388
#    0.15 0.1 -1.734601 -0.4626235 400  0.05 0.8998785
#    0.15 0.1 -1.734601 -0.4626235 450  0.05 0.9302222
#    0.15 0.1 -1.734601 -0.4626235 500  0.05 0.9518824
#
#  URL: http://psychstat.org/logistic

#To plot the power curve:
plot(res)

#To calculate the required sample size given power and effect size:
wp.logistic(n = NULL, p0 = 0.15, p1 = 0.1, alpha = 0.05,
              power = 0.8, family = "normal", parameter = c(0,1))
#  Power for logistic regression
#
#      p0  p1     beta0      beta1        n alpha power
#    0.15 0.1 -1.734601 -0.4626235 298.9207  0.05   0.8
#
#  URL: http://psychstat.org/logistic

Description

This function is for SEM power analysis based on the chi-square difference test.

Usage

wp.mc.chisq.diff(full.model.pop, full.model, 
reduced.model, N=100, R=1000, alpha=0.05)

Arguments

Value

An object of the power analysis.

References

Demidenko, E. (2007). Sample size determination for logistic regression revisited. Statistics in medicine, 26(18), 3385-3397.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

set.seed(20220722)

full.model.pop <-'
y1 ~ 0.4*x
y2 ~ 0.5*x + 0.2*y1
y3 ~ 0.4*x
y4 ~ 0.4*y1 + 0.4*y2 + 0.4*y3
y1 ~~ 0.84*y1
y2 ~~ 0.61*y2
y3 ~~ 0.84*y3
y4 ~~ 0.27*y4
'

full.model <-'
y1 ~ x
y2 ~ x + y1
y3 ~ x
y4 ~ y1 + y2 + y3
'

reduced.model <-'
y1 ~ x
y2 ~ x 
y3 ~ x
y4 ~ y1 + y3
'

wp.mc.chisq.diff(full.model.pop, full.model, reduced.model)

Description

Structural equation modeling (SEM) is a multivariate technique used to analyze relationships among observed and latent variables. It can be viewed as a combination of factor analysis and multivariate regression analysis. A mediation model can be viewed as a SEM model. Funtions wp.sem.chisq and wp.sem.rmsea provide anlytical solutions of power analysis for SEM. Function wp.mediation provides anlytical solutions of power analysis for a simple mediatoin model. This function provides a solution based on Monte Carlo simulation (see Zhang, 2014). If the model is a mediation, Sobel test is used for the mediation / indirect effects. The solution is extended from the general framework for power analysis for complex mediation models using Monte Carlo simulation in Mplus (Muthén & Muthén, 2011) proposed by Thoemmes et al. (2010). We extended the framework in two ways. First, the method allows the specification of nonnormal data in the Monte Carlo simulation and can thereby reflect more closely practical data collection. Second, the function wp.mc.sem.basic of a free, open-source R package, WebPower, is developed to ease power anlysis for mediation models using the proposed method.

Usage

wp.mc.sem.basic(model, indirect = NULL, nobs = 100, nrep = 1000,
  alpha = 0.95, skewness = NULL, kurtosis = NULL, ovnames = NULL,
  se = "default", estimator = "default", parallel = "no",
  ncore = Sys.getenv("NUMBER_OF_PROCESSORS"), cl = NULL, ...)

Arguments

Value

An object of the power analysis. The power for all parameters in the model as well as the indirect effects if specified.

References

MacCallum, R. C., Browne, M. W., & Sugawara, H. M. (1996). Power analysis and determination of sample size for covariance structure modeling. Psychological methods, 1(2), 130.

Rosseel, Y. (2012). Lavaan: An R package for structural equation modeling and more. Version 0.5–12 (BETA). Ghent, Belgium: Ghent University.

Satorra, A., & Saris, W. E. (1985). Power of the likelihood ratio test in covariance structure analysis. Psychometrika, 50(1), 83-90.

Thoemmes, F., MacKinnon, D. P., & Reiser, M. R. (2010). Power analysis for complex mediational designs using Monte Carlo methods. Structural Equation Modeling, 17(3), 510-534.

Zhang, Z. (2014). Monte Carlo based statistical power analysis for mediation models: Methods and software. Behavior research methods, 46(4), 1184-1198.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

set.seed(20220722)

#To calculate power for mediation based on Monte Carlo simulation when Sobel test is used:
#To specify the model
demo ="
		y ~ cp*x + start(0)*x + b*m + start(0.39)* m
		m ~ a*x + start(0.39)*x
		x ~~ start(1)*x
		m ~~ start(1)*m
		y ~~ start(1)*y
	"
#To specify the indirect effects
mediation = "
		ab := a*b
		abc:= a*b + cp
" 
#To calculate power for mediation using regular standard errors
## change nrep to at least 1,000 in real data analysis
sobel.regular = wp.mc.sem.basic(model=demo, indirect=mediation, nobs=100, nrep=10,
       skewness=c(0, 0, 1.3), kurtosis=c(0,0,10), ovnames=c("x","m","y"))

#To calculate power for mediation using robust standard errors
sobel.robust = wp.mc.sem.basic(model=demo, indirect=mediation, nobs=100, nrep=10,
   skewness=c(0, 0, 1.3), kurtosis=c(0,0,10), ovnames=c("x","m","y"), se="robust")

#To print the power for mediation based on Sobel test using regular standard errors:
summary(sobel.regular)
#    Basic information:
#    
#      Esimation method                                  ML
#      Standard error                              standard
#      Number of requested replications                1000
#      Number of successful replications               1000
#    
#                           True  Estimate      MSE      SD     Power Coverage
#    Regressions:
#      y ~
#        x        (cp)      0.000    0.003    0.106    0.107    0.045    0.955
#        m         (b)      0.390    0.387    0.099    0.113    0.965    0.919
#      m ~
#        x         (a)      0.390    0.389    0.100    0.101    0.976    0.953
#    Variances:
#        x                  1.000    0.995    0.141    0.139    1.000    0.936
#        m                  1.000    0.981    0.139    0.137    1.000    0.923
#        y                  1.000    0.968    0.137    0.330    1.000    0.560
#    
#    Indirect/Mediation effects:
#        ab                 0.152    0.150    0.056    0.060    0.886    0.928
#        abc                0.152    0.153    0.106    0.109    0.305    0.948

#To print the power analysis results for mediation based on Sobel test using robust standard errors:
summary(sobel.robust)
#    Basic information:
#    
#      Esimation method                                  ML
#      Standard error                            robust.sem
#      Number of requested replications                1000
#      Number of successful replications               1000
#    
#                           True  Estimate      MSE      SD     Power Coverage
#    Regressions:
#      y ~
#        x        (cp)      0.000   -0.003    0.106    0.113    0.055    0.945
#        m         (b)      0.390    0.398    0.111    0.119    0.972    0.927
#      m ~
#        x         (a)      0.390    0.389    0.099    0.101    0.974    0.939
#    
#    Intercepts:
#        y                  0.000    0.000    0.100    0.104    0.058    0.942
#        m                  0.000    0.000    0.100    0.105    0.054    0.946
#        x                  0.000   -0.004    0.100    0.104    0.066    0.934
#    
#    Variances:
#        x                  1.000    0.991    0.138    0.140    1.000    0.930
#        m                  1.000    0.976    0.135    0.135    1.000    0.915
#        y                  1.000    1.002    0.281    0.365    0.981    0.805
#    
#    Indirect/Mediation effects:
#        ab                 0.152    0.156    0.060    0.064    0.870    0.900
#        abc                0.152    0.153    0.108    0.117    0.303    0.936

Description

Structural equation modeling (SEM) is a multivariate technique used to analyze relationships among observed and latent variables. It can be viewed as a combination of factor analysis and multivariate regression analysis. A mediation model can be viewed as a SEM model. Funtions wp.sem.chisq and wp.sem.rmsea provide anlytical solutions of power analysis for SEM. Function wp.mediation provides anlytical solutions of power analysis for a simple mediatoin model. This function provides a solution based on Monte Carlo simulation (see Zhang, 2014) and a bootstrap method for testing the indirect /mediation effects. The solution is extended from the general framework for power analysis for complex mediation models using Monte Carlo simulation in Mplus (Muthén & Muthén, 2011) proposed by Thoemmes et al. (2010). We extended the framework in three ways. First, we proposes a general method to conduct power analysis for mediation models based on the bootstrap method. The method is still based on Monte Carlo simulation but uses the bootstrap method to test mediation effects. Second, the method allows the specification of nonnormal data in the Monte Carlo simulation and can thereby reflect more closely practical data collection. Third, the function wp.mc.sem.boot of a free, open-source R package, WebPower, is developed to ease power anlysis for mediation models using the proposed method.

Usage

wp.mc.sem.boot(model, indirect = NULL, nobs = 100, nrep = 1000,
  nboot = 1000, alpha = 0.95, skewness = NULL, kurtosis = NULL,
  ovnames = NULL, se = "default", estimator = "default",
  parallel = "no", ncore = Sys.getenv("NUMBER_OF_PROCESSORS"), cl = NULL,
  ...)

Arguments

Value

An object of the power analysis. The power for all parameters in the model as well as the indirect effects if specified.

References

Rosseel, Y. (2012). Lavaan: An R package for structural equation modeling and more. Version 0.5–12 (BETA). Ghent, Belgium: Ghent University.

Thoemmes, F., MacKinnon, D. P., & Reiser, M. R. (2010). Power analysis for complex mediational designs using Monte Carlo methods. Structural Equation Modeling, 17(3), 510-534.

Zhang, Z. (2014). Monte Carlo based statistical power analysis for mediation models: Methods and software. Behavior research methods, 46(4), 1184-1198.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

set.seed(20220722)

#To specify the model
demo ="
		y ~ cp*x + start(0)*x + b*m + start(0.39)* m
		m ~ a*x + start(0.39)*x
		x ~~ start(1)*x
		m ~~ start(1)*m
		y ~~ start(1)*y
	"
#To specify the indirect effects
mediation = "
		ab := a*b
		abc:= a*b + cp
" 
#Power for mediation based on MC method when bootstrap method is used to test the effects:
# change nrep and nboot to at least 1,000 in real analysis

mediation.boot = wp.mc.sem.boot(model=demo, indirect=mediation, nobs=100,
                               nrep=10, nboot=10,
             skewness=c(0, 0, 1.3), kurtosis=c(0,0,10), ovnames=c("x","m","y"))

#To print the power analysis results
summary(mediation.boot)

#Example: Power for Simple Mediation Analysis
ex1model <- "
		math ~ c*ME + start(0)*ME + b*HE + start(0.39)*HE
		HE ~ a*ME + start(0.39)*ME
"

indirect <- "ab:=a*b"

# change nrep and nboot to at least 1,000 in real analysis
boot.normal <- wp.mc.sem.boot(ex1model,indirect, 50, nrep=10, nboot=10)
summary(boot.normal) 

boot.non.normal <- wp.mc.sem.boot(ex1model,indirect, 100, nrep=10, nboot=10,
                              skewness=c(-0.3, -0.7, 1.3),
                              kurtosis=c(1.5, 0, 5), ovnames=c('ME','HE','math'))
summary(boot.non.normal) 

#Example: Multiple Group Mediation Analysis (Moderated Mediation)
ex3model <- "
		y ~ start(c(0.283, 0.283))*x + c(c1,c2)*x + start(c(0.36, 0.14))*m + c(b1,b2)*m
		m ~ start(c(0.721, 0.721))*x + c(a1,a2)*x
		m =~ c(1,1)*m1 + start(c(0.8, 0.8))*m2 + start(c(0.8, 0.8))*m3
		x ~~ start(c(0.25, 0.25))*x
		y ~~ start(c(0.81, 0.95))*y
		m ~~ start(c(0.87, 0.87))*m
		m1 ~~ start(c(0.36, 0.36))*m1
		m2 ~~ start(c(0.36, 0.36))*m2
		m3 ~~ start(c(0.36, 0.36))*m3
"

# med1 and med2 are the mediation effect for group1 and group2, respectively.
indirect <- "
		med1 := a1*b1
		med2 := a2*b2
		diffmed := a1*b1 - a2*b2
"

# change nrep and nboot to at least 1,000 in real analysis
bootstrap <- wp.mc.sem.boot(ex3model, indirect, nobs=c(400,200),
                              nrep=10, nboot=10)
summary(bootstrap) 

#Example: A Longitudinal Mediation Model
ex4model <- "
x2 ~ start(.9)*x1 + x*x1
x3 ~ start(.9)*x2 + x*x2
m2 ~ start(.3)*x1 + a*x1 + start(.3)*m1 + m*m1
m3 ~ start(.3)*x2 + a*x2 + start(.3)*m2 + m*m2
y2 ~ start(.3)*m1 + b*m1 + start(.7)*y1 + y*y1
y3 ~ start(.3)*m2 + b*m2 + start(.7)*y2 + y*y2 + start(0)*x1 + c*x1
x1 ~~ start(.37)*m1
x1 ~~ start(.27)*y1
y1 ~~ start(.2278)*m1
x2 ~~ start(.19)*x2
x3 ~~ start(.19)*x3
m2 ~~ start(.7534)*m2
m3 ~~ start(.7534)*m3
y2 ~~ start(.3243)*y2
y3 ~~ start(.3243)*y3
"

indirect <- "ab := a*b"

# change nrep and nboot to at least 1,000 in real analysis
set.seed(10)
bootstrap <- wp.mc.sem.boot(ex4model, indirect, nobs=500, nrep=10, nboot=10)
summary(bootstrap)

Description

A power curve is useful to graphically display how power changes with sample size (e.g., Zhang & Wang). This function is to generate a power curve for SEM based on Monte Carlo simulation, either using Sobel test or bootstrap method to test the indirect / mediation effects if applicable.

Usage

wp.mc.sem.power.curve(model, indirect = NULL, nobs = 100, type = "basic",
  nrep = 1000, nboot = 1000, alpha = 0.95, skewness = NULL,
  kurtosis = NULL, ovnames = NULL, se = "default",
  estimator = "default", parallel = "no",
  ncore = Sys.getenv("NUMBER_OF_PROCESSORS"), cl = NULL, ...)

Arguments

Value

An object of the power analysis. The power for all parameters in the model as well as the indirect effects if specified.

References

Rosseel, Y. (2012). Lavaan: An R package for structural equation modeling and more. Version 0.5–12 (BETA). Ghent, Belgium: Ghent University.

Thoemmes, F., MacKinnon, D. P., & Reiser, M. R. (2010). Power analysis for complex mediational designs using Monte Carlo methods. Structural Equation Modeling, 17(3), 510-534.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

set.seed(20220722)

#To specify the model
ex2model ="
		ept ~ start(0.4)*hvltt + b*hvltt + start(0)*age + start(0)*edu + start(2)*R
		hvltt ~ start(-0.35)*age + a*age +c*edu + start(0.5)*edu
		R ~ start(-0.06)*age + start(0.2)*edu
		R =~ 1*ws + start(0.8)*ls + start(0.5)*lt
		age ~~ start(30)*age
		edu ~~ start(8)*edu
		age ~~ start(-2.8)*edu
		hvltt ~~ start(23)*hvltt
		R ~~ start(14)*R
		ws ~~ start(3)*ws
		ls ~~ start(3)*ls
		lt ~~ start(3)*lt
		ept ~~ start(3)*ept
	"
#To specify the indirect effects
indirect = "ind1 := a*b + c*b" 
nobs <- seq(100, 2000, by =200)
#To calculate power curve:
# change nrep and nboot to at least 1,000 in real analysis
power.curve = wp.mc.sem.power.curve(model=ex2model, indirect=indirect,
                              nobs=nobs, type='boot',
							  nrep=10, nboot=10)

Description

Power analysis for t-test based on Monte Carlo simulation

Usage

wp.mc.t(n = NULL, R0 = 1e+05, R1 = 1000, mu0 = 0, mu1 = 0, 
sd = 1, skewness = 0, kurtosis = 3, alpha = 0.05, 
type = c("two.sample", "one.sample", "paired"), 
alternative = c("two.sided", "less", "greater"))

Arguments

References

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

set.seed(20220722)
########## Chapter 16. Monte Carlo t-test #############
wp.mc.t(n=20 , mu0=0, mu1=0.5, sd=1, skewness=0, 
kurtosis=3, type = c("one.sample"), alternative = c("two.sided"))

wp.mc.t(n=40 , mu0=0, mu1=0.3, sd=1, skewness=1, 
kurtosis=6, type = c("paired"), alternative = c("greater"))

wp.mc.t(n=c(15, 15), mu1=c(0.2, 0.5), sd=c(0.2, 0.5), 
skewness=c(1, 2), kurtosis=c(4, 6), type = c("two.sample"), alternative = c("less"))

Description

This function is for mediation models. Mediation models can be used to investigate the underlying mechanisms related to why an input variable x influences an output variable y (e.g., Hayes, 2013; MacKinnon, 2008). The mediation effect is calculated as a*b, where a is the path coefficent from the predictor x to the mediator m, and b is the path coefficent from the mediator m to the outcome variable y. Sobel test statistic (Sobel, 1982) is used to test whether the mediation effect is significantly different from zero.

Usage

wp.mediation(n = NULL, power = NULL, a = 0.5, b = 0.5, varx = 1,
  vary = 1, varm = 1, alpha = 0.05, interval = NULL)

Arguments

Value

An object of the power analysis.

References

Hayes, A. F. (2013). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach. Guilford Press.

MacKinnon, D. P. (2008). Introduction to statistical mediation analysis. Routledge.

Sobel, M. E. (1982). Asymptotic confidence intervals for indirect effects in structural equation models. Sociological methodology, 13, 290-312.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

#To calculate the statistical power given sample size and effect size:
wp.mediation(n = 100, power = NULL, a = 0.5, b = 0.5,
                  varx = 1, vary = 1, varm = 1, alpha = 0.05)
#    Power for simple mediation
#
#        n     power   a   b varx varm vary alpha
#      100 0.9337271 0.5 0.5    1    1    1  0.05
#  
#    URL: http://psychstat.org/mediation

#To generate a power curve given a sequence of sample sizes:
res <- wp.mediation(n = seq(50,100,5), power = NULL, a = 0.5, b = 0.5,
                            varx = 1, vary = 1, varm = 1, alpha = 0.05)
res
#    Power for simple mediation
#  
#        n     power   a   b varx varm vary alpha
#       50 0.6877704 0.5 0.5    1    1    1  0.05
#       55 0.7287681 0.5 0.5    1    1    1  0.05
#       60 0.7652593 0.5 0.5    1    1    1  0.05
#       65 0.7975459 0.5 0.5    1    1    1  0.05
#       70 0.8259584 0.5 0.5    1    1    1  0.05
#       75 0.8508388 0.5 0.5    1    1    1  0.05
#       80 0.8725282 0.5 0.5    1    1    1  0.05
#       85 0.8913577 0.5 0.5    1    1    1  0.05
#       90 0.9076417 0.5 0.5    1    1    1  0.05
#       95 0.9216744 0.5 0.5    1    1    1  0.05
#      100 0.9337271 0.5 0.5    1    1    1  0.05
#  
#    URL: http://psychstat.org/mediation

#To plot the power curve:
plot(res)

#To calculate the required sample size given power and effect size:
wp.mediation(n = NULL, power = 0.9, a = 0.5, b = 0.5,
                varx = 1, vary = 1, varm = 1, alpha = 0.05)
#    Power for simple mediation
#
#             n power   a   b varx varm vary alpha
#      87.56182   0.9 0.5 0.5    1    1    1  0.05
#  
#    URL: http://psychstat.org/mediation

#To calculate the minimum detectable effect size of one coefficent given power and sample size:
wp.mediation(n = 100, power = 0.9, a = NULL, b = 0.5,
               varx = 1, vary = 1, varm = 1, alpha = 0.05)
#    Power for simple mediation
#  
#                     n power         a   b varx varm vary alpha
#      100   0.9 0.7335197 0.5    1    1    1  0.05
#  
#    URL: http://psychstat.org/mediation

Description

The two functions are adapted from the R package longpower by Michael C. Donohue. More will be added later.

Usage

wp.mmrm(N = NULL, Ra = NULL, ra = NULL, sigmaa = NULL, 
Rb = NULL, rb = NULL, sigmab = NULL, lambda = 1, 
delta = NULL, alpha = 0.05, power = NULL, 
alternative = c("two.sided", "one.sided"))
wp.mmrm.ar1(N = NULL, rho = NULL, ra = NULL, sigmaa = NULL, 
rb = NULL, sigmab = NULL, lambda = 1, times = 1:length(ra), 
delta = NULL, alpha = 0.05, power = NULL, 
alternative = c("two.sided", "one.sided"))

Arguments

References

Lu, K., Luo, X., Chen, P.-Y. (2008) Sample size estimation for repeated measures analysis in randomized clinical trials with missing data. International Journal of Biostatistics, 4, (1)

Description

power analysis of model 14 in Introduction to Mediation, Moderation, and Conditional Process Analysis. Powers are obtained through either the percentile bootstrap method or the Monte Carlo method. The conditional indirect effect value is a1(b1 + b2w); the index of moderated mediation is a1b2.

Usage

wp.modmed.m14(
  a1,
  cp,
  b1,
  d1,
  b2,
  sige12,
  sige22,
  n,
  sigx_w,
  sigx2 = 1,
  sigw2 = 1,
  nrep_power = 1000,
  alpha = 0.05,
  b = 1000,
  nb = n,
  w_value = 0,
  power_method = "product",
  simulation_method = "percentile",
  ncore = 1,
  pop.cov = NULL,
  mu = NULL,
  varnames = c("y", "x", "w", "m", "mw")
)

Arguments

Value

power of indirect effect, direct effect, moderation, and the index of moderated mediation

Examples

test = wp.modmed.m14(a1 = 0.2, cp = 0.2, b1 = 0.5, d1 = 0.5, b2 = 0.2, sigx2 = 1,
                    sigw2 = 1, sige12 = 1, sige22 = 1, sigx_w = 0.5, n = 50,
                    w_value = 0.5, simulation_method = "MC",
                    nrep_power = 1000, alpha = 0.05, b = 1000, ncore = 1)
print(test)

Description

power analysis of model 15 in Introduction to Mediation, Moderation, and Conditional Process Analysis. Powers are obtained through either the percentile bootstrap method or the Monte Carlo method. The conditional indirect effect value is a1(b1 + b2w); the conditional direct effect value is cp + d2w; the index of moderated mediation is c2b1.

Usage

wp.modmed.m15(
  a1,
  cp,
  b1,
  b2,
  d1,
  d2,
  sige12,
  sige22,
  sigx_w,
  n,
  sigx2 = 1,
  sigw2 = 1,
  nrep_power = 1000,
  alpha = 0.05,
  b = 1000,
  nb = n,
  w_value = 0,
  power_method = "product",
  simulation_method = "percentile",
  ncore = 1,
  pop.cov = NULL,
  mu = NULL,
  varnames = c("y", "x", "w", "m", "xw", "mw")
)

Arguments

Value

power of indirect effect, direct effect, moderation, and the index of moderated mediation

Examples

test = wp.modmed.m15(a1 = 0.6, cp = 0.2, b1 = 0.3, b2 = 0.2, d1 = 0.2, d2 = 0.1,
                     sigx2 = 1, sigw2 = 1, sige12 = 1, sige22 = 1, sigx_w = 0.4,
                     w_value = 0.3, simulation_method = "MC",
                     n = 50, nrep_power = 1000, b = 1000, alpha = 0.05, ncore = 1)
print(test)

Description

power analysis of model 58 in Introduction to Mediation, Moderation, and Conditional Process Analysis. Powers are obtained through either the percentile bootstrap method or the Monte Carlo method. The conditional indirect effect value is (a1 + c2w)(b1 + b2w); the index of moderated mediation cannot be applied here since the conditional indirect effect is not linear.

Usage

wp.modmed.m58(
  c1,
  a1,
  c2,
  d1,
  b1,
  b2,
  cp,
  sige12,
  sige22,
  sigx_w,
  n,
  sigx2 = 1,
  sigw2 = 1,
  nrep_power = 1000,
  alpha = 0.05,
  b = 1000,
  nb = n,
  w_value = 0,
  power_method = "product",
  ncore = 1,
  simulation_method = "percentile",
  pop.cov = NULL,
  mu = NULL,
  varnames = c("x", "w", "m", "xw", "mw", "y")
)

Arguments

Value

power of indirect effect, direct effect, and moderation

Examples

test = wp.modmed.m58(c1 = 0.2, a1 = 0.2, c2 = 0.1, b2 = 0.1,
     b1 = 0.2, cp = 0.2, d1 = 0.2, w_value = 0.3, simulation_method = "MC",
     sigx2 = 1, sigw2 = 1, sige12 = 1, sige22 = 1, sigx_w = 0.5,
     n = 50, nrep_power = 1000, b = 1000, alpha = 0.05, ncore = 1)
print(test)

Description

power analysis of model 7 in Introduction to Mediation, Moderation, and Conditional Process Analysis. Powers are obtained through either the percentile bootstrap method or the Monte Carlo method. The conditional indirect effect value is (a1+c2w)b1; the index of moderated mediation is c2b1.

Usage

wp.modmed.m7(
  a1,
  cp,
  b1,
  c1,
  c2,
  sige12,
  sige22,
  sigx_w,
  n,
  sigx2 = 1,
  sigw2 = 1,
  nrep_power = 1000,
  alpha = 0.05,
  b = 1000,
  nb = n,
  w_value = 0,
  power_method = "product",
  simulation_method = "percentile",
  ncore = 1,
  pop.cov = NULL,
  mu = NULL,
  varnames = c("y", "x", "w", "m", "xw")
)

Arguments

Value

power of indirect effect, direct effect, moderation, and the index of moderated mediation

Examples

# usage of wp.modmed.m7
test = wp.modmed.m7(a1 = 0.39, cp = 0.2, b1 = 0.3, c1 = 0.39,
          c2 = 0.2, sigx2 = 1, sigw2 = 1, sige12 = 1,
         sige22 = 1, sigx_w = 0.5, n = 50, b = 1000, nrep_power = 1000, simulation_method = "MC",
         alpha = 0.05, ncore = 1)
print(test)

Description

power analysis of model 8 in Introduction to Mediation, Moderation, and Conditional Process Analysis. Powers are obtained through either the percentile bootstrap method or the Monte Carlo method. The conditional indirect effect value is (a1 + c2w)b1; the conditional direct effect value is cp + d2w; the index of moderated mediation is c2b1.

Usage

wp.modmed.m8(
  a1,
  cp,
  b1,
  c1,
  d1,
  c2,
  d2,
  sige12,
  sige22,
  sigx_w,
  n,
  sigx2 = 1,
  sigw2 = 1,
  nrep_power = 1000,
  alpha = 0.05,
  b = 1000,
  nb = n,
  w_value = 0,
  power_method = "product",
  simulation_method = "percentile",
  ncore = 1,
  pop.cov = NULL,
  mu = NULL,
  varnames = c("y", "x", "w", "m", "xw")
)

Arguments

Value

power of indirect effect, direct effect, moderation, and the index of moderated mediation

Examples

# usage of wp.modmed.m8
test = wp.modmed.m8(a1 = 0.2, cp = 0.2, b1 = 0.2,
                   c1 = 0.2, d1=0.2, c2 = 0.2, d2 = 0.2,
                   sigx2 = 1, sigw2 = 1, sige12 = 1, sige22 = 1, sigx_w = 0.5,
                   simulation_method = "MC", 
                   n = 50, nrep_power = 1000, alpha = 0.05, ncore = 1)
print(test)

Description

Multisite randomized trials (MRT) are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a control arm (Liu, 2013). The data from MRT can be analyzed in a two-level hierarchical linear model, where the indicator variable for treatment assignment is included in first level. If a study contains multiple treatments, then multiple indicators will be used. This function is for designs with 2 arms (i.e., a treatment and a control). Three types of tests are considered in the function: (1) The "main" type tests treatment main effect; (2) The "site" type tests the variance of cluster/site means; and (3) The "variance" type tests variance of treatment effects. Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013).

Usage

wp.mrt2arm(n = NULL, f = NULL, J = NULL, tau00 = NULL, tau11 = NULL,
  sg2 = NULL, power = NULL, alpha = 0.05, alternative = c("two.sided",
  "one.sided"), type = c("main", "site", "variance"), interval = NULL)

Arguments

Value

An object of the power analysis.

References

Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.

Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

#To calculate the statistical power given sample size and effect size:
#For main effect
wp.mrt2arm(n = 45, f = 0.5, J = 20, tau11 = 0.5,
            sg2 = 1.25, alpha = 0.05, power = NULL)
#  Power analysis for Multileve model Multisite randomized trials with 2 arms
#
#      J  n   f tau11  sg2     power alpha
#      20 45 0.5   0.5 1.25 0.8583253  0.05
#
#  NOTE: n is the number of subjects per cluster
#  URL: http://psychstat.org/mrt2arm

#For variance of treament effect
wp.mrt2arm(n = 45, f = 0.5, J = 20, tau11 = 0.5,
               sg2 = 1.25, alpha = 0.05, power = NULL, type = "variance")
#  Power analysis for Multileve model Multisite randomized trials with 2 arms
#
#     J  n   f tau11  sg2     power alpha
#    20 45 0.5   0.5 1.25 0.9987823  0.05
#
#  NOTE: n is the number of subjects per cluster
#  URL: http://psychstat.org/mrt2arm

#For testing site variablity
res<- wp.mrt2arm(n = 45, J = 20, tau00 = 0.5,
                 sg2 = 1.25, alpha = 0.05, power = NULL, type = "site")
#  Power analysis for Multileve model Multisite randomized trials with 2 arms
#
#     J  n tau00  sg2     power alpha
#    20 45   0.5 1.25 0.9999999  0.05
#
#  NOTE: n is the number of subjects per cluster
#  URL: http://psychstat.org/mrt2arm

#To generate a power curve given a sequence of sample sizes:
res <- wp.mrt2arm(n = seq(10,50,5), f = 0.5, J = 20, tau11 = 0.5,
                          sg2 = 1.25, alpha = 0.05, power = NULL)
#  Power analysis for Multileve model Multisite randomized trials with 2 arms
#
#      J  n   f tau11  sg2     power alpha
#     20 10 0.5   0.5 1.25 0.6599499  0.05
#     20 15 0.5   0.5 1.25 0.7383281  0.05
#     20 20 0.5   0.5 1.25 0.7818294  0.05
#     20 25 0.5   0.5 1.25 0.8090084  0.05
#     20 30 0.5   0.5 1.25 0.8274288  0.05
#     20 35 0.5   0.5 1.25 0.8406659  0.05
#     20 40 0.5   0.5 1.25 0.8506049  0.05
#     20 45 0.5   0.5 1.25 0.8583253  0.05
#     20 50 0.5   0.5 1.25 0.8644864  0.05
#
#  NOTE: n is the number of subjects per cluster
#  URL: http://psychstat.org/mrt2arm

#To plot the power curve:
plot(res) 

#To calculate the required sample size given power and effect size:
wp.mrt2arm(n = NULL, f = 0.5, J = 20, tau11 = 0.5,
                      sg2 = 1.25, alpha = 0.05, power = 0.8)
#  Power analysis for Multileve model Multisite randomized trials with 2 arms
#
#     J        n   f tau11  sg2 power alpha
#    20 23.10086 0.5   0.5 1.25   0.8  0.05
#
#  NOTE: n is the number of subjects per cluster
#  URL: http://psychstat.org/mrt2arm

Description

Multisite randomized trials (MRT) are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a control arm (Liu, 2013). The data from MRT can be analyzed in a two-level hierarchical linear model, where the indicator variable for treatment assignment is included in first level. If a study contains multiple treatments, then multiple indicators will be used. This function is for designs with 3 arms (i.e., two treatments and a control). Three types of tests are considered in the function: (1) The "main" type tests treatment main effect; (2) The "treatment" type tests the difference between the two treatments; and (3) The "omnibus" type tests whether the three arms are all equivalent. Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013).

Usage

wp.mrt3arm(n = NULL, f1 = NULL, f2 = NULL, J = NULL, tau = NULL,
  sg2 = NULL, power = NULL, alpha = 0.05, alternative = c("two.sided",
  "one.sided"), type = c("main", "treatment", "omnibus"), interval = NULL)

Arguments

Value

An object of the power analysis.

References

Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.

Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

#To calculate the statistical power given sample size and effect size:
#For main effect
wp.mrt3arm(n = 30, f1 = 0.43, J = 20, tau = 0.4,
               sg2 = 2.25, alpha = 0.05, power = NULL)
#    Multisite randomized trials with 3 arms
#
#       J  n   f1 tau  sg2     power alpha
#      20 30 0.43 0.4 2.25 0.8066964  0.05
#  
#    NOTE: n is the number of subjects per cluster
#    URL: http://psychstat.org/mrt3arm

#For tesing difference between effects
wp.mrt3arm(n = 30, f2 = 0.2, J = 20, tau = 0.4, sg2 = 2.25,
                      alpha = 0.05, power = NULL, type="treatment")
#    Multisite randomized trials with 3 arms
#
#      J  n  f2 tau  sg2     power alpha
#      20 30 0.2 0.4 2.25 0.2070712  0.05
# 
#    NOTE: n is the number of subjects per cluster
#    URL: http://psychstat.org/mrt3arm

#For testing site variablity
wp.mrt3arm(n = 30, f1=0.43, f2 = 0.2, J = 20, tau = 0.4, sg2 = 2.25,
                          alpha = 0.05, power = NULL, type="omnibus")
#    Multisite randomized trials with 3 arms
#
#       J  n   f1  f2 tau  sg2     power alpha
#      20 30 0.43 0.2 0.4 2.25 0.7950757  0.05
#  
#    NOTE: n is the number of subjects per cluster
#    URL: http://psychstat.org/mrt3arm

#To generate a power curve given a sequence of numbers of sites/clusters:
res <- wp.mrt3arm(n = 30, f2 = 0.2, J = seq(20,120,10), tau = 0.4,
            sg2 = 2.25, alpha = 0.05, power = NULL, type="treatment")
res
#    Multisite randomized trials with 3 arms
#
#       J  n  f2 tau  sg2     power alpha
#      20 30 0.2 0.4 2.25 0.2070712  0.05
#      30 30 0.2 0.4 2.25 0.2953799  0.05
#      40 30 0.2 0.4 2.25 0.3804554  0.05
#      50 30 0.2 0.4 2.25 0.4603091  0.05
#      60 30 0.2 0.4 2.25 0.5337417  0.05
#      70 30 0.2 0.4 2.25 0.6001544  0.05
#      80 30 0.2 0.4 2.25 0.6593902  0.05
#      90 30 0.2 0.4 2.25 0.7116052  0.05
#     100 30 0.2 0.4 2.25 0.7571648  0.05
#     110 30 0.2 0.4 2.25 0.7965644  0.05
#     120 30 0.2 0.4 2.25 0.8303690  0.05
# 
#    NOTE: n is the number of subjects per cluster
#    URL: http://psychstat.org/mrt3arm

#To plot the power curve:
plot(res, "J", "power")

#To calculate the required sample size given power and effect size:
wp.mrt3arm(n = NULL, f1 = 0.43, J = 20, tau = 0.4,
                sg2 = 2.25, alpha = 0.05, power = 0.8)
#    Multisite randomized trials with 3 arms
#
#       J        n   f1 tau  sg2 power alpha
#      20 28.61907 0.43 0.4 2.25   0.8  0.05
#  
#    NOTE: n is the number of subjects per cluster
#    URL: http://psychstat.org/mrt3arm

Description

This function is for Poisson regression models. Poisson regression is a type of generalized linear models where the outcomes are usually count data. Here, Maximum likelihood methods is used to estimate the model parameters. The estimated regression coefficent is assumed to follow a normal distribution. A Wald test is used to test the mean difference between the estimated parameter and the null parameter (tipically the null hypothesis assumes it equals 0). The procedure introduced by Demidenko (2007) is adopted here for computing the statistical power.

Usage

wp.poisson(n = NULL, exp0 = NULL, exp1 = NULL, alpha = 0.05,
  power = NULL, alternative = c("two.sided", "less", "greater"),
  family = c("Bernoulli", "exponential", "lognormal", "normal", "Poisson",
  "uniform"), parameter = NULL, subdivisions=200L,
  i.method=c("numerical", "MC"), mc.iter=20000)

Arguments

Value

An object of the power analysis.

References

Demidenko, E. (2007). Sample size determination for logistic regression revisited. Statistics in medicine, 26(18), 3385-3397.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

#To calculate the statistical power given sample size and effect size:
wp.poisson(n = 4406, exp0 = 2.798, exp1 = 0.8938, alpha = 0.05,
                 power = NULL, family = "Bernoulli", parameter = 0.53)
#  Power for Poisson regression
#
#       n     power alpha  exp0   exp1    beta0      beta1 paremeter
#    4406 0.9999789  0.05 2.798 0.8938 1.028905 -0.1122732      0.53
#
#  URL: http://psychstat.org/poisson

#To generate a power curve given a sequence of sample sizes:
res <- wp.poisson(n = seq(800, 1500, 100), exp0 = 2.798, exp1 = 0.8938,
      alpha = 0.05, power = NULL, family = "Bernoulli", parameter = 0.53)
res
#  Power for Poisson regression
#
#       n     power alpha  exp0   exp1    beta0      beta1 paremeter
#     800 0.7324097  0.05 2.798 0.8938 1.028905 -0.1122732      0.53
#     900 0.7813088  0.05 2.798 0.8938 1.028905 -0.1122732      0.53
#    1000 0.8224254  0.05 2.798 0.8938 1.028905 -0.1122732      0.53
#    1100 0.8566618  0.05 2.798 0.8938 1.028905 -0.1122732      0.53
#    1200 0.8849241  0.05 2.798 0.8938 1.028905 -0.1122732      0.53
#    1300 0.9080755  0.05 2.798 0.8938 1.028905 -0.1122732      0.53
#    1400 0.9269092  0.05 2.798 0.8938 1.028905 -0.1122732      0.53
#    1500 0.9421344  0.05 2.798 0.8938 1.028905 -0.1122732      0.53
#
#  URL: http://psychstat.org/poisson

#To plot the power curve:
plot(res) 

#To calculate the required sample size given power and effect size:
wp.poisson(n = NULL, exp0 = 2.798, exp1 = 0.8938, alpha = 0.05,
            power = 0.8, family = "Bernoulli", parameter = 0.53)
#  Power for Poisson regression
#
#           n power alpha  exp0   exp1    beta0      beta1 paremeter
#    943.2628   0.8  0.05 2.798 0.8938 1.028905 -0.1122732      0.53
#
#  URL: http://psychstat.org/poisson

Description

This function is used to extract population value table for parameters form an power analysis object for SEM based on Monte Carlo methods (class = 'power').

Usage

wp.popPar(object)

Arguments

Value

Population value table of parameters from the input object of power analysis.

Description

Tests of proportions are a technique used to compare proportions of success or agreement in one or two samples. The one-sample test of proportion tests the null proportion of success, usually 0.5. The two-sample test of proportions tests the null hypothesis that the two samples are drawn from populations with the same proportion of success. A z-test is used to evaluate whether the given difference in proportions is statistical significantly different from the null hypothesis. The power calculation is based on the arcsine transformation of the proportion (see Cohen, 1988, p.548).

Usage

wp.prop(h = NULL, n1 = NULL, n2 = NULL, alpha = 0.05, power = NULL,
  type = c("1p", "2p", "2p2n"), alternative = c("two.sided", "less",
  "greater"))

Arguments

Value

An object of the power analysis.

References

Cohen, J. (1992). A power primer. Psychological bulletin, 112(1), 155.

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed). Hillsdale, NJ: Lawrence Erlbaum Associates.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

#To calculate the power for two groups of proportion with unequal sample size:
wp.prop(h=0.52,n1=35,n2=50,alternative="greater",type="2p2n")
#  Power for two-sample proportion (unequal n)
#
#       h n1 n2 alpha     power
#    0.52 35 50  0.05 0.7625743
#
#  NOTE: Sample size for each group
#  URL: http://psychstat.org/prop2p2n

#To calculate the power curve with a sequence of sample sizes:
res <- wp.prop(h=0.52,n1=seq(10,100,10),alternative="greater",type="1p")
res
#  Power for one-sample proportion test
#
#       h   n alpha     power
#    0.52  10  0.05 0.4998128
#    0.52  20  0.05 0.7519557
#    0.52  30  0.05 0.8855706
#    0.52  40  0.05 0.9499031
#    0.52  50  0.05 0.9789283
#    0.52  60  0.05 0.9914150
#    0.52  70  0.05 0.9965928
#    0.52  80  0.05 0.9986772
#    0.52  90  0.05 0.9994960
#    0.52 100  0.05 0.9998111
#
#  URL: http://psychstat.org/prop

#To plot the power curve:
plot(res, type='b')

#To estimate the sample size with a given power:
wp.prop(h=0.52,n1=NULL,power=0.8,alternative="greater",type="1p")
#  Power for one-sample proportion test
#
#       h        n alpha power
#    0.52 22.86449  0.05   0.8
#
#  URL: http://psychstat.org/prop

#To estimate the minimum detectable effect size with a given power:
wp.prop(h=NULL,n1=35,power=0.8,alternative="greater",type="1p")
#  Power for one-sample proportion test
#
#            h  n alpha power
#    0.4202907 35  0.05   0.8
#
#  URL: http://psychstat.org/prop

#To calculate the power curve with a sequence of effect sizes:
wp.prop(h=seq(0.1, 0.8, 0.1),n1=100,alternative="greater",type="1p")
#  Power for one-sample proportion test
#
#      h   n alpha     power
#    0.1 100  0.05 0.2595110
#    0.2 100  0.05 0.6387600
#    0.3 100  0.05 0.9123145
#    0.4 100  0.05 0.9907423
#    0.5 100  0.05 0.9996034
#    0.6 100  0.05 0.9999934
#    0.7 100  0.05 1.0000000
#    0.8 100  0.05 1.0000000
#
#  URL: http://psychstat.org/prop

Description

This function is for power analysis for regression models. Regression is a statistical technique for examining the relationship between one or more independent variables (or predictors) and one dependent variable (or the outcome). Regression provides an F-statistic that can be formulated using the ratio between variation in the outcome variable that is explained by the predictors and the unexplained variation (Cohen, 1988)). The test statistc can also be experessed in terms of caomparison between Full and Reduced models (Maxwell & Delaney, 2003).

Usage

wp.regression(n = NULL, p1 = NULL, p2 = 0, f2 = NULL, alpha = 0.05,
  power = NULL, type=c("regular", "Cohen"))

Arguments

Value

An object of the power analysis

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed). Hillsdale, NJ: Lawrence Erlbaum Associates.

Maxwell, S. E., & Delaney, H. D. (2004). Designing experiments and analyzing data: A model comparison perspective (Vol. 1). Psychology Press.

Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

Examples

#To calculate the statistical power given sample size and effect size:
wp.regression(n = 100, p1 = 3, f2 = 0.1, alpha = 0.05, power = NULL)
#  Power for multiple regression
#
#      n p1 p2  f2 alpha     power
#    100  3  0 0.1  0.05 0.7420463
#
#  URL: http://psychstat.org/regression

#To generate a power curve given a sequence of sample sizes:
res <- wp.regression(n = seq(50,300,50), p1 = 3, f2 = 0.1, 
                                 alpha = 0.05, power = NULL)
res
#  Power for multiple regression
#
#      n p1 p2  f2 alpha     power
#    50  3  0 0.1  0.05 0.4077879
#    100  3  0 0.1  0.05 0.7420463
#    150  3  0 0.1  0.05 0.9092082
#    200  3  0 0.1  0.05 0.9724593
#    250  3  0 0.1  0.05 0.9925216
#    300  3  0 0.1  0.05 0.9981375
#
#  URL: http://psychstat.org/regression

#To plot the power curve:
plot(res) 

#To calculate the required sample size given power and effect size:
wp.regression(n = NULL, p1 = 3, f2 = 0.1, alpha = 0.05, power = 0.8)
#  Power for multiple regression
#
#           n p1 p2  f2 alpha power
#    113.0103  3  0 0.1  0.05   0.8
#
#  URL: http://psychstat.org/regression

#The statistical power given sample size and effect size when controling two predictors:
wp.regression(n = 100, p1 = 3, p2 = 2, f2 = 0.1429, alpha = 0.05, power = NULL)
#  Power for multiple regression
#
#      n p1 p2     f2 alpha     power
#    100  3  2 0.1429  0.05 0.9594695
#
#  URL: http://psychstat.org/regression

# To generate a power curve given a sequence of effect sizes:
res <- wp.regression(n = 50, p1 = 3, f2 = seq(0.05,0.5,0.05),
                                  alpha = 0.05, power = NULL)
res
#  Power for multiple regression
#
#     n p1 p2   f2 alpha     power
#    50  3  0 0.05  0.05 0.2164842
#    50  3  0 0.10  0.05 0.4077879
#    50  3  0 0.15  0.05 0.5821296
#    50  3  0 0.20  0.05 0.7210141
#    50  3  0 0.25  0.05 0.8220164
#    50  3  0 0.30  0.05 0.8906954
#    50  3  0 0.35  0.05 0.9350154
#    50  3  0 0.40  0.05 0.9624324
#    50  3  0 0.45  0.05 0.9788077
#    50  3  0 0.50  0.05 0.9883012
#
#  URL: http://psychstat.org/regression