Conversion from Probabilities to Cohen's w

Helper function to convert (multinomial or product-multinomial) probabilities to Cohen's w.

Usage

probs.to.w(prob.matrix,
           null.prob.matrix = NULL,
           verbose = TRUE)

Arguments

prob.matrix

a vector or matrix of cell probabilities under alternative hypothesis

null.prob.matrix

a vector or matrix of cell probabilities under null hypothesis. Calculated automatically when prob.matrix is specified. The default can be overwritten by the user via providing a vector of the same size or matrix of the same dimensions as prob.matrix

verbose

logical; whether the output should be printed on the console. TRUE by default.

Value

w

Cohen's w effect size. It can be any of Cohen's W, Phi coefficient, Cramer's V. Phi coefficient is defined as sqrt(X2/n) and Cramer's V is defined as sqrt(X2/(n*v)) where v is min(nrow - 1, ncol - 1) and X2 is the chi-square statistic.

df

degrees of freedom.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.

Examples

  # ---------------------------------------------------------#
  # Example 1: Cohen's W                                     #
  # goodness-of-fit test for 1 x k or k x 1 table            #
  # How many subjects are needed to claim that               #
  # girls choose STEM related majors less than males?       #
  # ---------------------------------------------------------#

  ## from https://www.aauw.org/resources/research/the-stem-gap/
  ## 28 percent of the  workforce in STEM field is women
  prob.vector <- c(0.28, 0.72)
  null.prob.vector <- c(0.50, 0.50)
  probs.to.w(prob.vector, null.prob.vector)
#>    w   df 
#> 0.44 1.00 

  power.chisq.gof(w = 0.44, df = 1,
                  alpha = 0.05, power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Chi-Square Test for Goodness-of-Fit or Independence
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim)   : P[i,j] = P0[i,j] for all (i,j) 
#>   H1 (Alt. Claim)   : P[i,j] != P0[i,j] for some (i,j)
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Total Sample Size      = 41  << 
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.196
#>   Statistical Power      = 0.804
#> 


  # ---------------------------------------------------------#
  # Example 2: Phi Coefficient (or Cramer's V or Cohen's W)  #
  # test of independence for 2 x 2 contingency tables        #
  # How many subjects are needed to claim that               #
  # girls are underdiagnosed with ADHD?                      #
  # ---------------------------------------------------------#

  ## from https://time.com/growing-up-with-adhd/
  ## 5.6 percent of girls and 13.2 percent of boys are diagnosed with ADHD
  prob.matrix <- rbind(c(0.056, 0.132),
                       c(0.944, 0.868))
  colnames(prob.matrix) <- c("Girl", "Boy")
  rownames(prob.matrix) <- c("ADHD", "No ADHD")
  prob.matrix
#>          Girl   Boy
#> ADHD    0.056 0.132
#> No ADHD 0.944 0.868

  probs.to.w(prob.matrix)
#>         w        df 
#> 0.1302134 1.0000000 

  power.chisq.gof(w = 0.1302134, df = 1,
                  alpha = 0.05, power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Chi-Square Test for Goodness-of-Fit or Independence
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim)   : P[i,j] = P0[i,j] for all (i,j) 
#>   H1 (Alt. Claim)   : P[i,j] != P0[i,j] for some (i,j)
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Total Sample Size      = 463  << 
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.200
#>   Statistical Power      = 0.8
#> 


  # --------------------------------------------------------#
  # Example 3: Cramer's V (or Cohen's W)                    #
  # test of independence for j x k contingency tables       #
  # How many subjects are needed to detect the relationship #
  # between depression severity and gender?                 #
  # --------------------------------------------------------#

  ## from https://doi.org/10.1016/j.jad.2019.11.121
  prob.matrix <- cbind(c(0.6759, 0.1559, 0.1281, 0.0323, 0.0078),
                       c(0.6771, 0.1519, 0.1368, 0.0241, 0.0101))
  rownames(prob.matrix) <- c("Normal", "Mild", "Moderate",
                             "Severe", "Extremely Severe")
  colnames(prob.matrix) <- c("Female", "Male")
  prob.matrix
#>                  Female   Male
#> Normal           0.6759 0.6771
#> Mild             0.1559 0.1519
#> Moderate         0.1281 0.1368
#> Severe           0.0323 0.0241
#> Extremely Severe 0.0078 0.0101

  probs.to.w(prob.matrix)
#>          w         df 
#> 0.03022008 4.00000000 

  power.chisq.gof(w = 0.03022008, df = 4,
                  alpha = 0.05, power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Chi-Square Test for Goodness-of-Fit or Independence
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim)   : P[i,j] = P0[i,j] for all (i,j) 
#>   H1 (Alt. Claim)   : P[i,j] != P0[i,j] for some (i,j)
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Total Sample Size      = 13069  << 
#>   Type 1 Error (alpha)   = 0.050
#>   Type 2 Error (beta)    = 0.200
#>   Statistical Power      = 0.8
#>