Multiple regression is the analysis behind a huge share of MS and PhD theses in management, education, psychology, economics and public health. It lets you ask how several predictors together explain an outcome, and how much each one contributes while the others are held constant. Once you understand the small worked example below, the large SPSS or R output tables in your own thesis will make far more sense.
The Model
With two predictors, the multiple regression equation is
$$ \hat{y} = b_0 + b_1x_1 + b_2x_2 $$
- \( b_0 \) is the intercept: the predicted outcome when both predictors are zero.
- \( b_1 \) is the expected change in \( y \) for a one-unit increase in \( x_1 \), holding \( x_2 \) constant.
- \( b_2 \) is the same for \( x_2 \), holding \( x_1 \) constant.
The coefficients are chosen by ordinary least squares (OLS), which minimises the sum of squared residuals \( \sum (y - \hat{y})^2 \). The phrase "holding constant" is what makes multiple regression more informative than a set of separate correlations.
Worked Example: Predicting Exam Marks
A researcher collects data from six university students: weekly study hours (\( x_1 \)), a motivation score from 1 to 10 (\( x_2 \)) and final exam marks (\( y \)). A real thesis needs a much larger sample; six cases keep the arithmetic visible.
| Student | Study hours \( x_1 \) | Motivation \( x_2 \) | Marks \( y \) |
|---|---|---|---|
| 1 | 2 | 4 | 48 |
| 2 | 4 | 7 | 60 |
| 3 | 5 | 3 | 52 |
| 4 | 6 | 6 | 62 |
| 5 | 8 | 5 | 63 |
| 6 | 11 | 5 | 75 |
Step 1: Means
\( \bar{x}_1 = 36/6 = 6 \), \( \bar{x}_2 = 30/6 = 5 \), \( \bar{y} = 360/6 = 60 \).
Step 2: Deviations From the Means
- \( x_1 - \bar{x}_1 \): −4, −2, −1, 0, 2, 5
- \( x_2 - \bar{x}_2 \): −1, 2, −2, 1, 0, 0
- \( y - \bar{y} \): −12, 0, −8, 2, 3, 15
Step 3: Sums of Squares and Cross-Products
- \( S_{11} = 16+4+1+0+4+25 = 50 \)
- \( S_{22} = 1+4+4+1+0+0 = 10 \)
- \( S_{12} = 4 - 4 + 2 + 0 + 0 + 0 = 2 \)
- \( S_{1y} = 48+0+8+0+6+75 = 137 \)
- \( S_{2y} = 12+0+16+2+0+0 = 30 \)
- \( S_{yy} = 144+0+64+4+9+225 = 446 \)
Step 4: Solve for the Slopes
For two predictors, the least-squares solution is
$$ b_1 = \frac{S_{22}S_{1y} - S_{12}S_{2y}}{S_{11}S_{22} - S_{12}^2}, \qquad b_2 = \frac{S_{11}S_{2y} - S_{12}S_{1y}}{S_{11}S_{22} - S_{12}^2} $$
The common denominator is \( 50 \times 10 - 2^2 = 496 \).
$$ b_1 = \frac{10(137) - 2(30)}{496} = \frac{1310}{496} = 2.641 $$
$$ b_2 = \frac{50(30) - 2(137)}{496} = \frac{1226}{496} = 2.472 $$
Step 5: The Intercept
$$ b_0 = \bar{y} - b_1\bar{x}_1 - b_2\bar{x}_2 = 60 - 2.641(6) - 2.472(5) = 60 - 15.847 - 12.359 = 31.79 $$
So the fitted equation is
$$ \hat{y} = 31.79 + 2.641x_1 + 2.472x_2 $$
Each extra weekly study hour is associated with about 2.6 more marks, and each extra motivation point with about 2.5 more marks, holding the other predictor constant. A student with 7 study hours and motivation 6 is predicted to score \( 31.79 + 18.49 + 14.83 = 65.11 \).
Step 6: R Squared
The regression (explained) sum of squares is
$$ SS_R = b_1S_{1y} + b_2S_{2y} = 2.641(137) + 2.472(30) = 435.99 $$
The residual sum of squares is \( SS_E = 446 - 435.99 = 10.01 \), so
$$ R^2 = \frac{435.99}{446} = 0.978 $$
The two predictors explain 97.8% of the variance in marks. Adjusted \( R^2 \), which penalises extra predictors, is
$$ R^2_{adj} = 1 - (1 - R^2)\frac{n-1}{n-k-1} = 1 - 0.022 \times \frac{5}{3} = 0.963 $$
Such high values are typical of tidy teaching data; in real social-science theses, \( R^2 \) between .20 and .50 is common and perfectly respectable.
Step 7: The Overall F Test
$$ F = \frac{SS_R/k}{SS_E/(n-k-1)} = \frac{435.99/2}{10.01/3} = \frac{218.0}{3.337} = 65.32 $$
With df = (2, 3), p = .003, so the model as a whole predicts marks significantly better than the mean alone.
Step 8: Individual Coefficients
Software gives the standard error of each slope. Here SE(\( b_1 \)) = 0.259 and SE(\( b_2 \)) = 0.580, so
- Study hours: \( t = 2.641/0.259 = 10.18 \), p = .002
- Motivation: \( t = 2.472/0.580 = 4.26 \), p = .024
Standardised coefficients (beta weights), which put predictors on the same scale, are \( \beta_1 = 0.88 \) and \( \beta_2 = 0.37 \). Study hours is the stronger predictor in this sample.
Assumptions You Must Check and Report
- Linearity: each predictor relates linearly to the outcome (scatterplots, residual plots).
- Independence of errors: check the Durbin–Watson statistic, ideally near 2.
- Homoscedasticity: residuals have constant spread across predicted values.
- Normality of residuals: histogram or P–P plot of standardised residuals.
- No serious multicollinearity: variance inflation factors (VIF) below about 5, or tolerance above 0.2. Here \( r_{12} = 0.09 \), so VIF is about 1.01.
- No influential outliers: Cook's distance below 1.
A common rule of thumb for sample size is \( N \ge 50 + 8k \) for testing the overall model, where \( k \) is the number of predictors.
Reporting in APA Style
A multiple regression was conducted to predict exam marks from weekly study hours and motivation. The model was significant, F(2, 3) = 65.32, p = .003, and explained 97.8% of the variance (R² = .978, adjusted R² = .963). Study hours (B = 2.64, β = .88, p = .002) and motivation (B = 2.47, β = .37, p = .024) were both significant positive predictors.
Software
- SPSS: Analyze → Regression → Linear; request Collinearity diagnostics and Durbin–Watson under Statistics.
- Stata:
regress marks hours motivation, thenestat vif. - Excel: Data Analysis ToolPak → Regression.
- R:
r
model <- lm(marks ~ hours + motivation, data = df)
summary(model)
car::vif(model)
Did You Know?
- The word "regression" comes from Francis Galton, who noticed in the 1880s that very tall parents tended to have children closer to average height, which he called "regression towards mediocrity".
- Adding any predictor, even a random one, can never decrease \( R^2 \), which is why adjusted \( R^2 \) exists.
- One-way ANOVA and the independent t-test are both special cases of regression with dummy variables.
Key Takeaways
- Each coefficient shows a predictor's effect while holding the others constant.
- \( R^2 \) gives explained variance; the F test checks the model overall; t tests check each predictor.
- Check linearity, independence, homoscedasticity, normal residuals and multicollinearity.
- Report F, df, p, \( R^2 \), and B, β and p for each predictor.
- Regression on survey data shows prediction, not proof of causation.
If your thesis needs a regression model built, checked and written up, Amathify's experts and PhD scholars can guide you step by step; contact us on WhatsApp or through the Ask a Question form.
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