Types of Tests

Quick Note on Confidence Intervals: -​ CIs cannot include 0: Tests where 0 means no difference (e.g., t-tests (independent/paired), ANOVA (one-way/repeated measures), Regression (linear, multiple), Wilcoxon Signed-Rank Test, Kruskal-Wallis Test, Correlation (Pearson’s or Spearman’s). -​ CIs cannot include 1: Tests where 1 means no effect or no association (e.g., OR, RR, logical regression, chi-square). Parametric Tests Nonparametric Tests Independent T-test Purpose: to compare means of 2 groups Mann-Whitney U Test Purpose: to compare distribution (not means) of 2 independent samples Assumptions: -​ Independent variable is dichotomous -​ The 2 groups are independent -​ Dependent variable is continuous -​ Dependent variable is normally distributed -​ Homogeneity of variance (variances are roughly equal) -​ The independent variable is nominal or categorical and contains 2 levels (you have 2 mutually exclusive groups of subjects) Assumptions: -​ Independent variable is dichotomous -​ The 2 groups are independent -​ Dependent variable is at least ordinal -​ Total sample size at least 8 (or more) DF = n1 + n2 - 2 -​ (reject null if t-statistic is equal or greater than the critical value) Paired T-test Purpose: to compare the means of 2 related/dependent groups (repeated measures tests) Wilcoxon Matched-Pairs Signed Rank Test Purpose: to compare the medians of 2 related/dependent groups when the dependent variable is not normally distributed Assumptions: -​ The independent variable is dichotomous -​ The 2 groups are dependent -​ The dependent variable is continuous -​ The dependent variable is normally distributed Assumptions: -​ The independent variable is dichotomous -​ The 2 groups are dependent -​ The dependent variable is ordinal, interval, or ratio -​ There are at least 5 pairs of measurements DF = n - 1 DF = n - 1 ANOVA Purpose: compare the means of 3 or more independent samples Kruskal-Wallis H-test Purpose: compare ranked scores of 3 or more independent samples that are not normally distributed Assumptions: -​ The independent variable has 3 or more categories -​ BUT: One-way ANOVA only has 2 groups -​ The groups are independent -​ Dependent variable is normally distributed -​ Dependent variable is continuous -​ Homogeneity of variance Assumptions: -​ The independent variable has 3 or more categories -​ The groups are independent -​ Dependent variable is not normally distributed -​ Dependent variable is at least ordinal Properties of F distribution: -​ The F-distribution is positively skewed toward the right with a range of 0 to infinity -​ The F value is + or 0 (no negative values) -​ The shape of the distribution depends on the degree of freedom of the denominator Notes: -​ Null hypothesis: all the means should be relatively equal -​ Alternative hypothesis: at least one pair from the 3+ groups is not equal -​ DF numerator = K (total # of groups) - 1 -​ DF denominator = N (total # of subjects) - 1 Post-HOC Tests: -​ All you need to know about post-HOC tests is that ANOVA will tell you whether or not there is a difference in the group means, but not where that difference is. You use post-HOC tests to find where that difference is -​ Used to determine which specific groups means are different after finding a significant F statistic -​ Eg. Scheffe, Bonferroni Notes: -​ Degrees of freedom are calculated by DF = (number of groups compared) - 1 -​ DF = K - 1 There is not exactly a perfect parametric equivalent to chi-square, but you may be able to do the Fisher's Exact Test, ANOVA, or Chi-Square Goodness of Fit test as a parametric alternative. -​ -​ -​ Fisher’s Exact test: small sample when observed in one cell <5 Yates Correction: more than 2 x 2 table (testing for independence and contingency) Goodness of Fit Test: determine if sample matches a population Chi-Square Test Purpose: compares expected to observed to see associations between 2 things Assumptions: -​ The data is frequency data -​ The measures are both nominal -​ There is an adequate sample size (>3) -​ The measures are independent DF = (rows -1) x (columns - 1) Characteristics: 1.​ The chi-square distribution is a family of curves based on the degrees of freedom 2.​ The chi-square distributions are positively skewed 3.​ All chi-square values are greater than or equal to zero 4.​ The total area under each distribution is equal to 1 Pearson’s r Purpose: describes the direction and strength of a linear relationship between two variables that are normally distributed Spearman’s p Purpose: describes the direction and strength of a linear relationship between two variables that are not normally distributed Assumptions: -​ Level of measurement: the two variables should be at the interval or ratio level -​ Linear relationship: there should be a linear relationship between the two variables -​ Normality: both variables should be roughly normally distributed -​ Random: the data should be randomly selected Assumptions: -​ At least one variable is ordinal and not normally distributed Notes: -​ Spearman and Pearson correlation coefficients describe the direction and strength of a linear relationship between two variables -​ -1 = perfect negative correlation -​ 0 = no correlation -​ +1 = perfect positive correlation Notes: -​ Spearman and Pearson correlation coefficients describe the direction and strength of a linear relationship between two variables -​ -1 = perfect negative correlation -​ 0 = no correlation -​ +1 = perfect positive correlation -​ The strength of the relationship or correlation is determined by the absolute value of the correlation coefficient -​ 0.9 = strong association -​ 0.5 = moderate association -​ 0.25 = weak association -​ -​ The strength of the relationship or correlation is determined by the absolute value of the correlation coefficient -​ <0.3 = weak association -​ 0.3-0.5 = moderate association -​ >0.5 = strong association This is the same as pearson’s strength but from the textbook ^ Why Pearson’s r is better: -​ If you square Pearson’s coefficient (r^2), it becomes the coefficient of determination. If you multiply it by 100 it gives you the percentage of variance *CORRELATION DOES NOT EQUAL CAUSATION* Coefficient of Determination (r-squared): -​ The proportion of variance in the dependent variable that can be explained by the independent variable -​ Aka: how much variance is common to each variable Absolute Risk, Absolute Risk Difference, Relative Risk, and Odds Ratios 1. Absolute Risk ●​ Definition: The probability or rate of an event occurring in a group. It is a straightforward measure of risk. ●​ Formula: ●​ Example:​ If 20 out of 1,000 smokers develop lung cancer, the absolute risk for smokers is: ○​ 20/1000 = 0.02 or 2% 2. Absolute Risk Difference/Reduction (ARD/ARR): ●​ Definition: Absolute Risk Difference is the difference in risk between two groups (e.g., exposed vs. unexposed). It quantifies the additional or reduced risk of an event in one group compared to another. ●​ Formula:​ ○​ It measures the actual difference in risk between two groups, helping to understand the impact of an exposure. ○​ ●​ Use: ARD is used to compare the risk of an event between two groups and understand how much more or less likely the event is in one group compared to the other. Example: ●​ Exposed Group: 100 people, 30 develop the disease. ○​ ●​ Unexposed Group: 150 people, 20 develop the disease. ○​ Now, calculate the Absolute Risk Difference: This means the exposed group has a 17% higher risk of developing the disease compared to the unexposed group. 3. Relative Risk (RR) ●​ Definition: A ratio comparing the risk of an event in one group to the risk in another group (e.g., exposed vs. unexposed). ●​ Formula: ●​ Example: ○​ If 20 out of 1,000 smokers develop lung cancer (risk = 2%), and 5 out of 1,000 non-smokers develop lung cancer (risk = 0.5%), the relative risk is: ●​ ○​ Interpretation: Smokers are 4 times more likely to develop lung cancer than non-smokers. 4. Odds Ratio (OR) ●​ Definition: The ratio of the odds of an event occurring in one group to the odds of it occurring in another group. Often used in case-control studies. ●​ Formula: ○​ Odds: The ratio of the probability of an event occurring to it not occurring: ●​ ●​ Example:​ Using the previous numbers: ○​ Smokers: ○​ Non-smokers: ○​ Odds Ratio: ●​ ○​ Interpretation: Smokers have 4.08 times the odds of developing lung cancer compared to non-smokers. 5. Attack Rate ●​ Definition: Attack rate is a measure used in outbreak investigations to describe the proportion of people who develop a disease or outcome in a defined population over a specific time period. It is typically used for acute outbreaks of infectious diseases (e.g., foodborne illness, flu). ●​ Formula: ​ ​ ●​ Expressed as a percentage. ●​ Key Features: ●​ Focus: Measures the risk of disease in a specific group during an outbreak. ●​ Time Frame: Short and specific (e.g., during an outbreak). ●​ Used to identify sources of exposure and determine the severity of an outbreak. ●​ Example of Attack Rate: ○​ At a picnic, 50 people ate contaminated food, and 20 of them developed food poisoning. ●​ This means the attack rate is 40%. Key Differences: ●​ Absolute Risk is a direct measure of likelihood for a group. ●​ Absolute Risk Difference compares the risks between two groups to understand how much the risk differs due to exposure or intervention. ●​ Relative Risk compares the likelihood between two groups. ●​ Odds Ratio uses odds rather than probabilities and is especially useful in retrospective studies. ●​ Attack Rate: Used for outbreaks, measures risk in a specific group.