Statistics Lecture 9: Correlation and ANOVA

Lecture 9: Correlation and ANOVA Tanmay’s Slides on ANOVA: -​ When comparing more than 2 groups: -​ If assumptions are met: ANOVA (parametric) -​ If assumptions are not met: Kruskal Wallis (non-parametric) -​ A significant p-value suggests that at least one group mean is significantly different that the others. -​ ANOVA (analysis of variance) is a parametric statistical test used to compare the means of 3 or more groups to see if there is a statistically significant difference between them -​ F-statistic is used as the test statistic -​ Types of ANOVA: -​ -​ -​ -​ -​ Assumptions of ANOVA: -​ The measurement of each population are normally distributed -​ The measurements of each population have equal variances -​ The samples are drawn randomly -​ The samples are independent of one another -​ There is a homogeneity of variance (use Levene’s test to check) 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 Post Hoc Test -​ Used to determine which specific groups means are different after finding a significant F statistic Kruskal Wallis: -​ The Kruskal Wallis H test is a non parametric statistical test used to determine if there are statistically significant differences between the medians of 3 or more independent groups -​ It is the alternative to the one-way ANOVA when the assumptions of normality and equal variances are not met -​ Degrees of freedom are calculated by DF = (number of groups compared) - 1 Anila’s Slides on Correlation: -​ Correlation coefficients are a measure of association between two numerical variables. It also indicates the direction and magnitude of the relationship between two variables -​ Assumptions: -​ The sample subjects should be independent and randomly selected -​ The level of measurement is interval/ratio -​ Linear relationship between variables -​ Homoscedasticity (equal or similar variances within the groups) -​ Selecting the best correlation test to use: -​ Relationship between categorical variables (nominal/ordinal) → Chi-square -​ At least one variable is ordinal and not normally distributed → Spearman correlation coefficient (p) -​ Two variables at the interval/ratio level and is normally distributed → Pearson correlation coefficient (r) -​ Spearman and Pearson correlation coefficients describe the direction and strength of a linear relationship between two variables -​ -1 indicates a perfect negative relationship -​ 0 indicates no relationship at all. (this means the variables are completely independent. The closer to 0, the weaker the relationship) -​ 1 indicates a perfect positive relationship -​ 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 -​ If you square Pearson’s coefficient (r2), it becomes the coefficient of determination. If you multiply it by 100 it gives you the percentage of variance -​ Correlation does not equal causation BACK TO REGULAR SLIDES ANOVA (F Ratio) -​ Comparing dependent variable mean scores among 3 or more groups -​ Are there differences between sample means larger than mere chance? -​ 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 Properties of the F Distribution: -​ The F distribution is positively skewed beginning at 0 -​ There are many F distributions and each one has a different shape. We specify a particular one by designating the degrees of freedom associated with S1 2 and S2 2, we denote these quantities by df1 (called numerator degrees of freedom) and df2 (called denominator degrees of freedom) Kruskal Wallis H Test -​ If assumption and ANOVA are not met: we use Kruskal Wallis H Test, the nonparametric equivalent of ANOVA. -​ This test compares the ranked scores of 3 or more independent samples that are not normally distributed 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 -​ Eg. Scheffe, Bonferroni Correlation -​ A measure of association between two variables -​ Indicates direction and magnitude of relationship between the two variables -​ With multiple variables a correlation matrix can be displayed -​ CORRELATION DOES NOT EQUAL CAUSATION -​ There are two correlation coefficients: -​ Pearson’s r (parametric) -​ Spearman’s rank (p) “rho” (nonparametric) -​ The correlation coefficient can have a value from -1 to +1 -​ -1 = perfect negative correlation -​ 0 = no correlation -​ +1 = perfect positive correlation -​ Strengths of association based on the correlation coefficients: -​ 0.9 = strong association -​ 0.5 = moderate association -​ 0.25 = weak association Covariance -​ A measure of relationship between variables -​ It is the variance shared between two variables -​ It allows us to know the strength of the correlation 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 Assumptions of Pearson Correlation -​ 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 Causation -​ Hills 9 criteria are used to help distinguish significant associations that indicate causation 1.​ Temporal relationship a.​ If factor “A” causes a disease, it must precede the occurrence of the disease b.​ Eg. you need exposure to COVID to get COVID 2.​ Strength of association (statistical tests) a.​ The stronger the association, the more likely A causes B (high r value) 3.​ Dose-response relationship a.​ An increasing amount of exposure increases the risk 4.​ Consistency a.​ If the association is consistent when the study is replicated in different settings/examples and methods, it is likely to be CAUSAL 5.​ Plausibility a.​ The association agrees with our understanding of underlying pathological processes 6.​ Consideration of alternate explanations a.​ When considering whether an association is CAUSAL, you have to determine other possible explanations and rule them out 7.​ Experiment a.​ The condition can be altered by an intervention 8.​ Specificity a.​ Means that a single cause produces a specific effect b.​ Weakest factor of all the criteria 9.​ Coherence a.​ The association is consistent with existing theory and knowledge … current state knowledge in a field