# C. Sean Burns: Notebook

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r:multiple-regression

#### Multiple Regression

The following example is based on:

Pedhazur, E. J. (1997). Multiple regression in behavioral research: Explanation and prediction (3rd ed.). Wadsworth.

From the chapter: “Elements of Multiple Regression Analysis: Two Independent Variables” (pp. 95-134).

Y  <- c(2,4,4,1,5,4,7,9,7,8,5,2,8,6,10,9,3,6,7,10)
X1 <- c(1,2,1,1,3,4,5,5,7,6,4,3,6,6,8,9,2,6,4,4)
X2 <- c(3,5,3,4,6,5,6,7,8,4,3,4,6,7,7,6,6,5,6,9)

where

2. X1 = verbal aptitude
3. X2 = achievement motivation

Objectives:

1. Find the constants in the multiple regression equation: $Y' = a + {b_1}{X_1} + {b_2}{X_2}$
2. Find the “proportion of variance 'accounted for,' that is $R^2_y.12$” (p. 99). I.e., how much does verbal aptitude and achievement motivation explain the variance in reading achievement.
3. Test for statistical significance of the whole model or test if the coefficients are statistically significant from zero.
4. Test the relative importance of X1 and X2 on Y.

(Note to self: Work through the section and address the above objectives and specific tests in more detail.)

mr.ex <- data.frame(Y,X1,X2)
fit.1 <- lm(Y ~ X1 + X2, data = mr.ex)
summary(fit.1)

Call:
lm(formula = Y ~ X1 + X2, data = mr.ex)

Residuals:
Min       1Q   Median       3Q      Max
-2.19708 -1.35307  0.04611  0.93246  2.32495

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.4707     1.1942  -0.394 0.698353
X1            0.7046     0.1753   4.021 0.000887 ***
X2            0.5919     0.2438   2.428 0.026580 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.514 on 17 degrees of freedom
Multiple R-squared:  0.7229,    Adjusted R-squared:  0.6903
F-statistic: 22.17 on 2 and 17 DF,  p-value: 1.83e-05