When we use t test to compare two sample means, the testing statistics of paired variables and that of independent variables are different. However, whether the two variables are dependent will not change the way of forming the testing statistics in the comparison of two sample proportions. Why?
(The attached picture is from the course material of Inferential Statistics, an online course by Duke.)
PS. I cannot imagine what we can do to adjust for the dependence of paired variables when comparing their proportions. But I still think there is something we should do as in the test for two sample means.
Sincere thanks to anyone who pays attention to this question!!
An interesting question, Joy.
If we were dealing with, say, a paired sample where the measures are the proportion of correct answers in quizes, then we might have data that looked something like the following, where G1 and G2 are group_1 and group_2, and G2-G1 is the difference in proportions:
G1 G2 G2-G1
10% 15% 5%
15% 20% 5%
20% 20% 0%
30% 40% 10%
70% 70% 0%
Mean 29% 33% 4%
If G1 and G2 are treated as independent samples, then the difference would be regarded as not different from zero (that is, not statistically significant), but if we recognise that G1 and G2 are paired (for example, same people at two different times, or husband and wife, etc.) then we can see that all proportions in G2 are either higher than, or the same as, G1.
I don't know how you'd do this with proportions only. Maybe some of our colleagues in RG can help.
My approach would be to transform the proportions into Z-scores (values for Z to denote location on a Standard Normal Distribution, with proportions as the area under the curve). (The NORM.INV function in Excel does this nicely.) Then you can run a regular paired t-test.
Z-scores for the above values are:
Z1 Z2 Z2-Z1
-1.282 -1.036 0.245
-1.036 -0.842 0.195
-0.842 -0.842 0.000
-0.524 -0.253 0.271
0.524 0.524 0.000
Mean -0.632 -0.490 0.142
and the paired t-test is as we would expect:
t-Test: Paired Two Sample for Means
Z1 Z2
Mean -0.632 -0.490
Variance 0.495 0.408
Observations 5 5
Pearson Correlation 0.985
Hypothesized Mean Difference 0
df 4
t Stat -2.397
P(T
Thanks to the answers from Hume and Ette. I realize that I did not make myself clear when using the term "comparison of two proportions". Let me try again with two specific examples, the first one involves proportions of two dependent categorical variables and the second one involves proportions of two independent categorical variables.
Example 1:
Suppose that a group of people are required to answer two questions, QA and QB. What I am interested in is If the rate of correctness for QA is higher than that of QB. (So the question of interest involves a comparison of two proportions, and two variables are categorical [either right or wrong for each observation].)
If we denote a correct answer as 1 and a wrong one as 0, the original observation would be like:
NO. QA QB
obs1 0 1
obs2 1 1
obs3 0 0
obs4 1 1
... ... ...
obsn 1 0
avg. a% b%
And I want to test the null hypothesis H0: population proportions a%=b%.
Example 2:
Suppose we have a group of female subjects and a group of male subjects. Both groups are asked to answer a question QC. And we are interested in whether the rate of correctness of female subjects is higher than that of male subjects.
The testing statistics for the two examples are both of a Z-statistics form. As is shown in the attached picture, the numerator of the testing statistics is the difference of the two proportions, and the denominator is formed with a pooled proportion (in example 1, p_pool would be the pooled rate of correctness of QA and QB; in example 2, p_pool would be the pooled rate of correctness of female group and male group.).
So the tests for example 1 and example 2 do not differ much (except for that we would be using n or n_female & n_male when forming the testing statistics).
My question is that since whether a certain subjects can answer correctly to QA and QB is correlated, should we do something to adjust for this correlation.
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