Module 6 - Correlation & Regression

Correlation

Correlation

relationship between two continuous variables, measure degree data points cluster around regression line

Line of Best Fit

show general pattern of relationship between dependent & independent variable

Correlation Coefficients

measure direction & magnitude of relationship between independent & dependent variable

Magnitude of Association

Numerical value of correlation coefficient, show strength of association, 0 = no linear association, -/+ 1 = perfect linear association

Direction of Association

If correlation coefficient positive or negative, show directionality of relationship (pos or neg correlation)

Pearson Correlation

measure degree of relationship between linear related variables

Assumptions of Pearson Correlation

scale measurements, normal distribution, 2 variables = paired, no outliers, linearity, homoscedasticity

Linearity

straight line relationship between 2 variables, as x increase -> y increase/decrease, check with scatter plot visualization

Homoscedasticity

equal spread of data around line of best fit, data = homoscedastic or heteroscedastic, check with scatter plot

r

correlation coefficient

Spearman's Rank Order Correlation

Non-parametric Pearson's correlation, ranks data to explore relationship between 2 variables

Assumptions of Spearman's Rank Order Correlation

scale/ordinal data, any distribution, linear relationship between variables

Intraclass Correlation Coefficient (ICC)

used to evaluate inter-rate reliability, test-retest reliability & intra-rater reliability; for data structured as groups (not pairs)

Inter-Rater Reliability

variation between >=2 raters, measuring same event

Test-Retest Reliability

variation in 2 measurements under same conditions

Intra-Rater Reliability

variation within 1 rater across >= 2 trails

Factors Impacting Correlations

Restricting Data Range (can sometimes be good), Heterogenous Samples, Outliers (alter correlation)

Clinical use of correlation

reliability to clinical assessments tools, impact medical decision making

Regression

also explore relationship between variables, how explanatory variable impact response variable

Response variable

variable you are predicting, outcome/dependent/y variable

explanatory variable

variable you use to predict, x/independent variable

residuals

distance observed y lies from regression line

epsilon

error term, represent residual

beta 0

y-intercept

beta 1

slope of regression line

Least Squares Method

regression - chooses values of y-intercept & slope that minimize sum of squared residuals

what does a lower squares value mean?

smaller difference between data points & line of best fit

Regression Assumptions

scale data, residuals of regression line are normally distributed, no outliers, linear relationship between 2 variables, data = homoscedastic

Least Squares Regression Model

Sum of residuals = 0, line of best fit passes through mean of x & mean of y

R square

coefficient of determination, amount of variation in y explained by x

interpret image coefficients

y-intercept = -77.283, slope = 3.33

Interpret model summary

R = simple correlation between variables, 57% of variation of y is explained by X

Interpret ANOVA results

model significantly predicts y

r squared characteristics

always positive, as approached 0 = low variation in Y determined by x, max = 1 (all variability in y is determined by x

Linear Regression

1 explanatory & 1 response variable, scale measurements

Multiple Linear Regressions

1 response variable, >1 explanatory variable, scale measurements

Logistic Regression

1 explanatory variable, 1 response variable (dichotomous)

statistical significance

probability of event occurring due to random chance

clinical significance

event/difference is meaningful for a clinical reason

biological significance

whether finding has biological relevance

Data reproducibility

ability to reproduce/replicate findings

Replication crisis

our current inability to reproduce scientific results

causes for replication crisis

publication bias, bad study design & power, questionable research practices

Questionable Research Practices

p-hacking, selective reporting, sampling bias, HARKing ( Hypothesis After Result is Known)

Solutions to replicability crisis

preregistration of studies, replication studies, open science, alternative statistical approaches, education