# Statistical Tests

**Comparison of means****â€‹**To compare the means between groupsis used.*t*-test**(AIIMS Mayâ€™08).**- When there is only one sample group and we take the means before and after interventions, it is called a
or a*matched*. This design is analyzed by using the*paired design***paired**(or, the matched-group*t*test*t*test). - When observations come from two separate or independent groups, the appropriate test is the
**two-sample independent â€“groups***t*test. - Similarly Z-test can be used instead of a T-test when the size of the sample is>100 and the population standard deviation is known.
- 95% Confidence Intervals (CI) are estimated as: Observed mean
__+__1.96 Standard Error of Mean (SEM). - To test for variance (standard deviation) between two independent groups,
**F-test**is used. - For comparison of three or more means the recommended approach is to use the
**analysis of variance**or**Anova.** - The above-mentioned tests assume an underlying gaussian distribution of the means, thus they are also called
**parametric tests.** - When the means have non-gaussian distribution then we use
**non-parametric tests,**which include**Wilcoxon Signed-Ranks tests**in a paired design, and for two independent groups we use the**Wilcoxon Rank-Sum Test (or Mann Whitney**Both tests compare the equality of medians rather than means.*U*).

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**Comparison of proportions**- Proportions are compared when the data is measured on a nominal or ordinal scale.
- The test, which is commonly used, is the
**chi square test.**It is a non-parametric test. - The test is done by constructing 2x2 contingency table and calculating the
*expected frequency*of the variable from the*observed frequency*. - The test can be applied in two or more independent groups and a paired group.
- For comparing proportions in paired-groups we use the
**McNemar test.** - When the expected frequencies are less than 5 (small sample size), the appropriate test is called
**Fisherâ€™s exact test.** - 95% CI is given by: Observed proportion
__+__1.96 SE of proportion.

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**Correlation & regression****â€‹**Correlation- It describes the relationship between two numerical variables on a scatter plot.
**Correlation coefficient (**is the measure of correlation. It varies from â€“1 to +1. These two values describe a perfect negative and positive linear relationship between two variables.*r*)*r*= 0 implies that thereâ€™s no linear relationship.- Squaring the correlation coefficient (
*r*^{2}) gives us the**coefficient of determination.**It tells us the percentage of variability in one variable, which can be accounted for by the other variable. - For two ordinal (or one ordinal and one numerical) variables the
**Spearman rank correlation**is used. It is also used when the numerical variables have a skewed distribution.

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**Regression****â€‹**Regression analysis is used to predict the value of one variable (dependent variable) from the knowledge of the other (independent variable).- The regression equation is:
**Y = a + b X**, where â€˜Yâ€™ = dependent variable, â€˜Xâ€™ = independent variable, â€˜aâ€™ = y-axis intercept and â€˜bâ€™ =**regression coefficient.**

**â€‹â€‹****Z Scores**

The location of any element in a normal distribution can be expressed in terms of how many standard deviations it lies above or below the mean of the distribution. This is the z-score of the element. If the element lies above the mean it will have a positive z score and vice versa.

**Z =**__X -____Î¼__

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