Difference in proportions

Packages

• pwr

Concepts

• sample size
• effect size
• significance level = P(Type I error) = probability of finding an effect that is not there
• power = 1 - P(Type II error) = probability of finding an effect that is there

With three you can calculate the fourth

Assumptions

Under normal circumstantes, there is a proportion of 35% of patients with the characteristic. An intervention or exposure that could increase by 10% difference (45% or more) could be consider as clinically significant.

Effect size

This difference in terms of effect size for a difference 0.45 to 0.35 is 0.2045252

ES.h(.45, .35)

[1] 0.2045252

Sample size calculations

Hence we calculate the required sample size for this effect size.

For equal groups:

10% difference

pwr.2p.test(h = ES.h(.45, .35)  ,  # this is the effect size

          n =  ,  # sample size to calculate

          sig.level = 0.05,  # 5% alpha value

          power = 0.8) # 80% power
        

pwr.2p.test(h = ES.h(.45, .35)  , 
            n =  , 
            sig.level = 0.05, 
            power = 0.8)

     Difference of proportion power calculation for binomial distribution (arcsine transformation) 

              h = 0.2045252
              n = 375.2691
      sig.level = 0.05
          power = 0.8
    alternative = two.sided

NOTE: same sample sizes

You need two groups of 376 patients each

20% difference

For a difference pf 20% (35 to 55)

pwr.2p.test(h = ES.h(.55, .35)  , 
            n =  , 
            sig.level = 0.05, 
            power = 0.8)

     Difference of proportion power calculation for binomial distribution (arcsine transformation) 

              h = 0.4048601
              n = 95.76939
      sig.level = 0.05
          power = 0.8
    alternative = two.sided

NOTE: same sample sizes

You need two groups of 96 patients each

For unequal groups

Assuming that in one group you have already 75 patients

20% difference

pwr.2p2n.test(h = 0.4048601 , 
              n1 = 75, 
              power = 0.8, 
              sig.level = 0.05)

     difference of proportion power calculation for binomial distribution (arcsine transformation) 

              h = 0.4048601
             n1 = 75
             n2 = 132.4474
      sig.level = 0.05
          power = 0.8
    alternative = two.sided

NOTE: different sample sizes

You need in the other group 133 patients