## Hotelling-Williams T-test (1)

Recently, I am trying to compare the performance of two measures. It turns out a problem of comparing two correlation coefficients $\rho_{12}$ and $\rho_{13}$, where the subscript 1 is denoting the observation group, 2 and 3 is denoting the measures. To be honest, I don't have any idea at the very beginning. Many thanks to my supvisor Dr. Dennis Cheung, he sent me a PPT about correlation coefficients, which Hotelling-Williams T test [Steiger] is also included.

The formula of Hotelling-Williams T test is here:

• N = Number of Observation
• $r_{12} =$ sample correlation between Observation and measure 2
• $r_{13} =$ sample correlation between Observation and measure 3
• $r_{23} =$ sample correlation between measures
• $|R| = 1 - r_{12}^2 - r_{13}^2 - r_{23}^2 + 2(r_{12})(r_{13})(r_{23})$
• $\bar{r} = (r_{12} + r_{13})/2$
• $\rho$ means population correlation and $r$ is denoting sample correlation

Hotelling-Williams T Test performs well in my hypothesis testing. It proofs that there is a significant difference between two measures, which explained the phenomenons I have observed. It is linear in my case, but I doubt that whether Hotelling-Williams T test appropriate for non-linear case, like log case . I found that in [crr] blog, there is a post about solving a similar problem --the correlations between the frequency measures and word processing time. Their post is very detailed and two more similar testing techniques are also introduced. One is the Vuong Test[Vuong, 1989], this test was suggested when dealing with a nonlinear problem, for example, the word processing time and log frequency. This will require we should use non-linear regression model. Vuong was suggested for this case for it based on a comparison of the log-likelihood. Another method is developed by Clarke (2007)[Clarke], he suspected that Vuong test is considered conservative for small N. However, after conducting a simulation test conducted by the [crr] blogger, they concluded that Hotelling-Williams T test is the best one and the latter is Vuong test. The Vuong test will be suggested unless the correlation between variables is very little.

The core idea about Hotelling-Williams T test is not clear yet, I will finish that in next post.

1. [crr]http://crr.ugent.be/archives/546
2. [Vuong] Vuong, Q.H. (1989): Likelihood Ratio Tests for Model Selection and non-nested Hypotheses. Econometrica, 57, 307-333.
3. [Clarke] Clarke, K.A. (2007). A Simple Distribution-Free Test for Nonnested Model Selection. Political Analysis, 15, 347-363.
4. [Steiger] Steiger, J.H. (1980), Tests for comparing elements of a correlation matrix, Psychological Bulletin, 87, 245-251.

Hotelling-Williams T 检验的公式如下：

• $r_{12} =$ correlation between Observation and measure 2
• $r_{13} =$ correlation between Observation and measure 3
• $r_{23} =$ correlation between measures
• N = Number of Observation
• $|R| = 1 - r_{12}^2 - r_{13}^2 - r_{23}^2 + 2(r_{12})(r_{13})(r_{23})$
• $\bar{r} = (r_{12} + r_{13})/2$

1. [crr]http://crr.ugent.be/archives/546
2. [Vuong] Vuong, Q.H. (1989): Likelihood Ratio Tests for Model Selection and non-nested Hypotheses. Econometrica, 57, 307-333.
3. [Clarke] Clarke, K.A. (2007). A Simple Distribution-Free Test for Nonnested Model Selection. Political Analysis, 15, 347-363.
4. [Steiger] Steiger, J.H. (1980), Tests for comparing elements of a correlation matrix, Psychological Bulletin, 87, 245-251.