Parametric vs. Nonparametric Tests

• Hypothesis tests in statistics are divided into parametric and nonparametric tests.
• They are used to infer information about population parameters or compare groups.

Parametric Tests

• Parametric tests assume a specific distribution, typically a normal distribution.
• Assumptions are made about the underlying population parameters, like mean and variance.
• T-tests, ANOVA, regression analysis, and parametric correlation tests are examples.
• More, powerful when assumptions are valid, but less robust if assumptions are violated.

Nonparametric Tests

• Nonparametric tests are distribution-free tests
• These do not assume an underlying population distribution.
• Tests are based on ranks or ordinal data.
• Used when data doesn't meet parametric test assumptions or when the true population distribution is unknown.
• Examples include Mann-Whitney U test, Wilcoxon signed-rank test, Kruskal-Wallis test, and Spearman's rank correlation test.
• Nonparametric tests are often more robust and flexible than parametric tests.
• Parametric tests are often preferred for their power and precise estimates when assumptions are met.
• Nonparametric tests are a reliable alternative if assumptions are violated or unsuitable for parametric tests.

Chi-Square Test

• The chi-square test determines if a significant relationship exists between two categorical variables.
• It tests if variables are independent or associated.
• Data is organized into a contingency table showing frequencies for each category.
• Observed frequencies are compared to expected frequencies assuming no relationship.
• A chi-square test is a nonparametric test, which avoids assumptions about the underlying data distribution.
• Based on differences between observed and expected frequencies, rather than specific numerical values.
• The chi-square statistic is compared to a critical value, or a p-value is calculated to determine statistical significance.
• The null hypothesis is rejected if the chi-square statistic exceeds the critical value or if the p-value is less than a significance level (e.g., 0.05).
• The chi-square test is used to assess relationships between categorical variables by comparing observed and expected frequencies.

Fisher's Exact Test

• Fisher's exact test determines the association between two categorical variables in a contingency table.
• Particularly useful when the sample size is small.
• It is an alternative to the chi-square test when it may not be valid or accurate due to small sample size or low expected cell counts.
• Most useful for analyzing contingency tables with small sample sizes, rare events, or sparse data.
• Commonly used in medical research, genetics, and other fields with limited sample sizes.
• Fisher's exact test is nonparametric and relies on assumptions about the data distribution or population parameters.
• Exact probabilities are calculated based on combinatorial calculations using the hypergeometric distribution.

T-Test

• The t-test is used to compare the means of two groups.
• Determines if a significant difference exists between them.
• Based on the t-statistic, measures the difference between sample means to variability within groups.
• The t-test is applicable when data approximately follows a normal distribution.
• Variances of the two groups are assumed to be equal, called homogeneity of variances.
• Named variations occur depending on the specific scenario.
• Independent two-sample t-test: Compares the means of two independent groups, assuming observations within each group are independent.
• Paired sample t-test: Compares the means of two related groups, such as before and after measurements, considering the paired nature of observations.
• It is a parametric test, making assumptions about the underlying population distribution, specifically normality.
• Assumes observations are independent and variances are equal in the independent two-sample t-test).
• Nonparametric alternatives such as the Mann-Whitney U test (for independent groups) or the Wilcoxon signed-rank test (for paired groups) can be used if assumptions are unmet.

Mann-Whitney U Test

• The Mann-Whitney U test compares the distributions of two independent groups.
• It assesses if the medians of the two groups differ significantly.
• Also known as the Wilcoxon rank-sum or Wilcoxon-Mann-Whitney test.
• Suitable for non-normal or skewed data, as it does not assume any specific distribution for the data.
• Ranking the combined data from both groups and comparing the sum of ranks assigned to each group is how it operates.
• The test calculates a U statistic, representing the probability of observing a randomly selected value from one group greater than a randomly selected value from the other group.

Kruskal-Wallis Test

• This is a nonparametric statistical test used to compare the distributions of three or more independent groups.
• It assesses significant differences in the medians of the groups and does not assume any specific data distribution.
• The test ranks combined data from all groups and calculates a test statistic based on the ranks.
• It follows a chi-square distribution with (k-1) degrees of freedom.

ANOVA

• Analysis of Variance or ANOVA is a statistical test used to compare the means of three or more groups.
• It assesses whether there are significant differences between the group means.
• Considering both within-group variability and between-group variability is part of the analysis.
• It addresses if there are any differences in the means of multiple groups, or are the observed differences due to random chance.
• Compares the variation between groups to the variation within groups, so If the variation between groups is significantly larger than the variation within groups suggests meaningful differences in population means.
• It is a parametric test because it assumes the data follow a normal distribution with equal variances.
• One-Way ANOVA: Used when there is one independent variable (factor) with three or more levels (groups). It compares the means across groups.
• Two-Way ANOVA: Used when there are two independent variables (factors) and their interaction, examining the main effects of each factor and the interaction effect between them.
• Repeated Measures ANOVA: Used when there are repeated measurements on individuals or units, analyzing the effects of factors.