# Statistical Tests

Describe the appropriate selection of non-parametric and parametric tests and tests that examine relationships (e.g. correlation, regression)

## Parametric Tests

Parametric tests are used when data is:

- Continuous and numerical
- Normally distributed
- Remember that due to the central limit theorem - large data sets (n > 100) are typically amenable to parametric analysis, as sample means will follow a normal distribution
- Non-normal data can be transformed so that they follow a normal distribution

- Samples are taken randomly
- Samples have the same variance
- Observations within the group are independent

Independent results are those when one value is not expected to influence another value.- A common example is repeated measures: when serial measures are taken from a patient or a hospital, the results cannot be treated as independent
**Paired tests**are used when two dependent samples are compared**Unpaired test**are used when two independent samples are compared

Tests may be one-tailed or two-tailed:

- A
**two-tailed test**evaluates whether the sample mean is significantly greater or less than the population mean - A
**one-tailed test**only evaluates the relationship in one direction

This doubles the power of the test to detect a difference, but should only be performed if there is a very good reason that the effect could only occur in one direction.

Common parametric tests include:

### Z test

Used to test whether the mean of a particular sample (x̄) differs from the population mean (μ) by random variation.

Assumptions:

- Large sample

n > 100. - Data is normally distributed
- Population standard deviation is known

### Student's T Test

This is a variant of the Z test, used when the population standard deviation is not known.

- The results from T test approximate the results of the Z test when n > 100

### F Test

Compares the ratio of variances ($$Var_1 \over Var_2$$) for two samples. If F deviates significantly from 1, then there is a significant difference in group variances.

### Analysis of Variance (ANOVA)

ANOVA tests for significant differences between means of multiple groups, in a more efficient manner than multiple comparisons (doing lots of T tests).

There are several types of ANOVA tests used in different situations.

## Non-Parametric Tests

Non-parametric tests are used when the assumptions for parametric tests are not met. Non-parametric tests:

- Do not assume the data follows any particular distribution

This is required when:- Non-normality is obvious

e.g. Multiple observations of 0 - Possible non-normality

Typically small sample sizes. - Data is ordinal

- Non-normality is obvious
- Are not as powerful as parametric tests (a larger sample size is required to achieve the same error rate)
- Are more broadly applicable than parametric tests as they do not require the same assumptions

Non-parametric tests still require that data:

- Is continuous or ordinal
- Within-group observations are independent
- Samples are taken randomly

In general, non-parametric tests;

- Take each result and rank them
- Calculations are then performed on each rank to find the test statistic

Common non-parametric tests include:

### Mann-Whitney U Test/Wilcoxon Rank Sum Test

Alternative to the unpaired T-test for non-parametric data.

Process:

- Data from both groups are combined, ordered, and given ranks
- Tied data are given identical ranks, where that rank is equal to the average rank of the tied observations

- The data are then separated into their original group
- Ranks in each group are added to give a test statistic for each group
- A statistical test is performed to see if the sum of ranks in one group is different to another

### Wilcoxon Signed Ranks Test

Alternative to the paired T-test for non-parametric data.

Process:

- As above (for the Wilcoxon Rank Sum Test), except absolute difference between paired observations are ranked

The sign (i.e. positive or negative) is preserved. - The sum of positive ranks is then compared with the sum of negative ranks
- If there is no difference between groups, we would expect the net value to be 0

## References

- Myles PS, Gin T. Statistical methods for anaesthesia and intensive care. 1st ed. Oxford: Butterworth-Heinemann, 2001.