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T-test & so on
Standard t-test
You can't use 'macro parameter character #' in math mode
t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$, where $\bar X$ are means, $s$ are variance, and $n$ are sample sizes. # t-test variations ###
Paired-data Tests#paired t-test (2 groups)
### Weighted t-test - = a variation of the standard t-test that takes into account the different variances or sample sizes of the groups being compared: $$ T = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{w_1 s_1^2}{n_1} + \frac{w_2 s_2^2}{n_2}}} $$, where $w1_$ and $w_2$ are the weights assigned to the variances or sample sizes of the two groups. - how to decide weights? - equal weights - inverse variance weights - inverse sample size weights - custom weights based on knowledge # t-test vs. z-test Both t-tests and z-tests can be used to test whether a population mean is equal to a particular value or whether the means from two different populations are equal. - the difference lies in test statistics: - z-test statistic follows a normal distribution - t-test statistic follows a student t-distribution (which involves estimating population variance) - Which to choose: - for small sample size (<30): if population variance is known → z-test, otherwise → t-test - for large sample size (>30): z-test works anyway because of central limit theorem - for very large sample size (>200): z-test works anyway because the t-distribution will resemble a normal distribution in this case # t-test vs. Cohen's d ### What is Cohen's d - a measure of effect size (i.e. the magnitude of the difference) used in statistics to quantify the difference between two group means in terms of standard deviation units. $$d = \frac{\bar x_1 - \bar x_2}{s_{pooled}}$$, where $$s_{pooled} = \sqrt{\frac{(n_1 - 1) s_1^2 + (n_2 - 1)s_2^2} {n_1 + n_2 -2}}$$and $s_1$ and $s_2$ are standard deviations. - provides a standardized measure of the effect size: typically, Cohen's d values of 0.2, 0.5, and 0.8 are considered small, medium, and large effect sizes ### Which to choose - you can use both! - The t-test and Cohen's d are complementary statistical measures used to assess and interpret differences between group means, with the t-test focusing on statistical significance and Cohen's d focusing on effect size. - While the t-test tells you whether the difference between the means is statistically significant, Cohen's d tells you how large or small that difference is in practical terms.
t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$, where $\bar X$ are means, $s$ are variance, and $n$ are sample sizes. # t-test variations ###
Paired-data Tests#paired t-test (2 groups)
### Weighted t-test - = a variation of the standard t-test that takes into account the different variances or sample sizes of the groups being compared: $$ T = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{w_1 s_1^2}{n_1} + \frac{w_2 s_2^2}{n_2}}} $$, where $w1_$ and $w_2$ are the weights assigned to the variances or sample sizes of the two groups. - how to decide weights? - equal weights - inverse variance weights - inverse sample size weights - custom weights based on knowledge # t-test vs. z-test Both t-tests and z-tests can be used to test whether a population mean is equal to a particular value or whether the means from two different populations are equal. - the difference lies in test statistics: - z-test statistic follows a normal distribution - t-test statistic follows a student t-distribution (which involves estimating population variance) - Which to choose: - for small sample size (<30): if population variance is known → z-test, otherwise → t-test - for large sample size (>30): z-test works anyway because of central limit theorem - for very large sample size (>200): z-test works anyway because the t-distribution will resemble a normal distribution in this case # t-test vs. Cohen's d ### What is Cohen's d - a measure of effect size (i.e. the magnitude of the difference) used in statistics to quantify the difference between two group means in terms of standard deviation units. $$d = \frac{\bar x_1 - \bar x_2}{s_{pooled}}$$, where $$s_{pooled} = \sqrt{\frac{(n_1 - 1) s_1^2 + (n_2 - 1)s_2^2} {n_1 + n_2 -2}}$$and $s_1$ and $s_2$ are standard deviations. - provides a standardized measure of the effect size: typically, Cohen's d values of 0.2, 0.5, and 0.8 are considered small, medium, and large effect sizes ### Which to choose - you can use both! - The t-test and Cohen's d are complementary statistical measures used to assess and interpret differences between group means, with the t-test focusing on statistical significance and Cohen's d focusing on effect size. - While the t-test tells you whether the difference between the means is statistically significant, Cohen's d tells you how large or small that difference is in practical terms.