Correlation & Linear Regression
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This document discusses various statistical analyses that can be used to analyze the relationship between variables: 1. Bivariate correlation measures the strength and direction of association between two variables using Pearson's correlation coefficient (r). A significant correlation is when p<0.05. 2. Linear regression analysis uses linear equations to model relationships between a dependent variable and one or more independent variables. It identifies outliers that do not follow patterns. 3. Multiple regression extends linear regression to multiple independent variables, allowing analysis of their collective influence on a dependent variable. It provides measures like R and R-squared of the model's accuracy and fit.
Correlation & Linear Regression
- 1. NYC Charter School Performance on the 2012-13 State Exams 1. Bivariate Correlations 2. Linear Regression Analysis 3. Multiple Regression Analysis
- 2. Bivariate Correlations Bivariate Correlation Paersone Correlation or Co-efficient of correlation Scale level of measurement
- 3. p<0.05 Significant Correlation Researcher can be 95% confident that the relationship between these two variables is not due to chance
- 4. Denoted -1 by r ≤ r ≤ +1 0 ------- ±0.3 No Relation ±0.3 ------- ±0.5 Weak Relation ±0.5 ------- ±0.8 Moderate Relation ±0.8 ------- ±1 Strong Relation
- 5. 1 is total positive correlation, 0 is no correlation, and −1 is negative correlation The closer the value is to -1 or +1, the stronger the association is between the variables
- 9. Outlier There should be no significant outliers. Outliers are simply single data points within your data that do not follow the usual pattern. The problem with outliers is that they can have a negative effect on the regression equation that is used to predict the value of the dependent (outcome) variable based on the independent (predictor) variable.
- 10. Multiple Regression: Model Sum a. R tells the reliability & mathematical relationship. 1. R Square (co-efficient of determination) tells the percentage of accuracy. 2. Also percentage of variation that can not be controlled i.e. 3. (1-R Square) i. Adjusted R2, It can be negative & always less than or equal to R ii. Adjusted R2 will be more useful only if the R2 is calculated based on a sample, not the entire population iii. Adjusted R2 increases only if the new term improves the model more than would be expected by chance
- 11. ANOVA ANOVA table tests whether the overall regression model is a good fit for the data. p<0.05 The table shows that the independent variables statistically significantly predict the dependent variable, F(3, 16) = 32.811, p < .0005 (i.e., the regression model is a good fit of the data)
- 12. Coefficients 𝑦 = −11.823 + 0.551𝑥1 + 0.104𝑥2 + 1.989𝑥3 How much the dependent variable varies with an independent variable , when all other independent variables are held constant. T value less than ±2 is not important Significant value of x