Multiple Regression in SPSS: A Worked Example

Step 1

The research question

People often argue about whether parents are to blame for their children's behaviour, or whether hanging around with the wrong crowd matters more. Fifty young people, their peers and their families were observed for one week, and every incident of anti-social behaviour (ASB) was logged. These are fictional data.

SPSS name	Role	What it measures
AntiSocial_Beh	Outcome (DV)	Number of incidences of anti-social behaviour displayed over 1 week
Parental_ASB	Predictor (IV)	Number of incidences of anti-social behaviour parents displayed over 1 week
Peer_ASB	Predictor (IV)	Number of incidences of anti-social behaviour peers displayed over 1 week
Substance_Use	Predictor (IV)	Number of times substances were misused over 1 week

Step 2

Running it in SPSS

Analyse → Regression → Linear…
Move AntiSocial_Beh into Dependent; move the three predictors into Independent(s). Leave Method on Enter.
Click Statistics… and tick the boxes below → Continue.
Click Plots… → tick Normal probability plot → Continue.
Click Save… → tick Cook's → Continue → OK.

Linear Regression

Dependent:

AntiSocial_Beh

Independent(s):

Parental_ASB

Peer_ASB

Substance_Use

Method: Enter

Linear Regression: Statistics

✓Estimates

✓Confidence intervals (95%)

✓Model fit

✓Descriptives

✓Collinearity diagnostics

Residuals

✓Durbin-Watson

Casewise diagnostics

Linear Regression: Plots

Standardized Residual Plots

Histogram

✓Normal probability plot

Linear Regression: Save

Distances

Mahalanobis

✓Cook's

Leverage values

Step 3

Checking assumptions

Before we trust the output, we check the assumptions: normality (the normal P-P plot), no influential cases (the three distance measures), no collinearity (tolerance and VIF, Step 8), and independent residuals (Durbin-Watson).

Normal P-P Plot of Regression Standardized Residual

Dependent Variable: AntiSocial_Beh

Unusual and influential cases: the three distance measures

Ticking Cook's in the Save dialog adds a new variable, COO_1, to the end of your Data View. The same dialog offers two related measures, Mahalanobis and Leverage values, which add MAH_1 and LEV_1. All three hunt for "unusual" cases, but each one asks a different question, so they make most sense side by side.

Data View (first 8 of 50 cases)

	AntiSocial_Beh	Parental_ASB	Peer_ASB	Substance_Use	MAH_1	LEV_1	COO_1
1	13	10	15	9	2.29	0.047	0.01291
2	12	10	10	13	0.15	0.003	0.00575
3	9	6	10	15	1.01	0.021	0.00217
4	15	12	17	21	3.02	0.062	0.00014
5	8	10	9	18	0.88	0.018	0.00145
6	15	9	14	22	1.89	0.038	0.01027
7	5	8	6	6	2.46	0.050	0.00101
8	4	5	6	15	2.05	0.042	0.00177

a. Distances added by SPSS: MAH_1 = Mahalanobis distance, LEV_1 = leverage (hat) value, COO_1 = Cook's distance.

Measure	The question it asks	Uses	Added as	Rule of thumb here
Leverage (hat value)	How far is this case's mix of predictor scores from everyone else's — i.e. how much pull could it have on the regression line?	Predictors only	LEV_1	The average across the sample is (k + 1) ÷ N = .08; anything above three times that, .24, has unusually strong pull.
Mahalanobis D²	How unusual is this case on the predictors, allowing for the way they correlate with each other — is it outside the range of our data?	Predictors only	MAH_1	Compare with chi-square with df = number of predictors (3): above 7.815 is unusual at p = .05.
Cook's distance	If I deleted this case, how much would the whole model move?	Predictors and outcome	COO_1	Above 1 = influential outlier; above 4 ÷ N = .08 is worth a look.

Check· Cook's distance: how much the model would change

Cook's distance answers the question that actually matters: if this one person's data were thrown away, how far would the fitted model move? It is built out of the other two ideas, so a case needs both a large residual (it sits off the line) and leverage (it sits away from the crowd) to score highly.

Above 1 = an influential outlier — the rule used in this lab. A stricter screening value of 4 ÷ N = 4 ÷ 50 = .08 is often used to decide which cases to inspect; being above it is a reason to look at the case, not a reason to delete it.

Our largest value is .263 (case 40). 6 cases are above the screening value, but every one is a long way below 1, so no case is unduly influencing the results. To find the maximum quickly, run Analyse → Descriptive Statistics → Descriptives on COO_1.

Durbin-Watson: are the residuals independent?

The statistic is printed in the Model Summary table (Step 6), but it is an assumption check, so it is dealt with here.

What it means· What 'independence of residuals' actually means

A residual is one person's own error: their actual ASB score minus the score the model predicted. Independence means those errors carry no pattern from one person to the next — knowing how wrong the model was about one young person tells you nothing about how wrong it was about another. Every error is a fresh, unrelated event.

When residuals are not independent we say they are autocorrelated (or show serial dependence): the error for one case predicts the error for the next, so two people's "mistakes" move together. The model is then tracking a pattern that has nothing to do with your predictors.

Where it goes wrong in practice: the same people measured on consecutive days (yesterday's mood carries into today's); related or clustered cases (siblings, friends in the same peer group, pupils in the same class); data collected in an order that means something, such as weeks or clinics; or a factor that drifts over time and was never measured, like a policy change or a seasonal effect.

What it means· How the statistic works — and why we want 1 to 3

Durbin-Watson compares each residual with the one immediately before it:

DW = Σ (residual − previous residual)² ÷ Σ residual²

The top line adds up the squares of those successive differences. If consecutive errors are almost the same (positive autocorrelation), the differences are tiny, the top line shrinks and DW falls towards 0. If consecutive errors swing from positive to negative (negative autocorrelation), the differences are large, the top line swells and DW climbs towards 4. If the errors are unrelated, the average squared difference between neighbours is about twice the average squared residual, and DW lands near 2.

So 2 is the neutral midpoint of a 0-to-4 scale, and "between 1 and 3" is simply the band either side of 2 in which the departure from independence is small enough to ignore: below 1, successive errors are similar enough to be worrying; above 3, they are alternating. Our value of 1.72 is 0.29 below 2 — slightly on the positive side, but well inside the band.

Strictly, the exact cut-off depends on N and the number of predictors, and SPSS prints the statistic without a significance value for it — which is why the lab uses the simple 1–3 rule. Note too that DW only tests the first-order pattern, each case against the one immediately before it, and that it reads the residuals in the order the cases sit in your file.

our value 1.72

01234

0 = strong positive autocorrelation2 = independent4 = strong negative autocorrelation

Step 4

Descriptive statistics

Descriptive Statistics

	Mean	Std. Deviation	N
AntiSocial_Beh	8.88	4.771	50
Parental_ASB	9.16	3.241	50
Peer_ASB	10.58	4.394	50
Substance_Use	14.32	6.153	50

Step 5

Correlations

		AntiSocial_Beh	Parental_ASB	Peer_ASB	Substance_Use
Pearson Correlation	AntiSocial_Beh	1.000	.426	.617	.373
	Parental_ASB	.426	1.000	.373	−.062
	Peer_ASB	.617	.373	1.000	.203
	Substance_Use	.373	−.062	.203	1.000
Sig. (1-tailed)	AntiSocial_Beh	.	.001	<.001	.004
	Parental_ASB	.001	.	.004	.334
	Peer_ASB	<.001	.004	.	.079
	Substance_Use	.004	.334	.079	.
N	AntiSocial_Beh	50	50	50	50
	Parental_ASB	50	50	50	50
	Peer_ASB	50	50	50	50
	Substance_Use	50	50	50	50

SPSS's regression correlation table gives one-tailed significance.

Step 6

Model summary

Model Summary

Model	R 1	R Square 2	Adjusted R Square 3	Std. Error of the Estimate 4	Durbin-Watson 5
1	.713ᵃ	.508	.476	3.452	1.715

a. Predictors: (Constant), Substance_Use, Parental_ASB, Peer_ASB
b. Dependent Variable: AntiSocial_Beh

Step 7

ANOVA: is the model significant?

ANOVA

Model		Sum of Squares	df 1	Mean Square	F 2	Sig. 3
1	Regression	567.050	3	189.017	15.860	.000ᵇ
	Residual	548.230	46	11.918
	Total	1115.280	49

a. Dependent Variable: AntiSocial_Beh
b. Predictors: (Constant), Substance_Use, Parental_ASB, Peer_ASB

Step 8

Coefficients: which predictors matter?

Coefficientsᵃ

Model	Unstandardized Coefficients		Standardized	t	Sig. 3	95.0% CI for B 4		Collinearity Statistics 5
Model	B 1	Std. Error	Beta 2	t	Sig. 3	Lower	Upper	Tolerance	VIF
(Constant)	-3.357	1.974		-1.701	.096	−7.330	0.616
Parental_ASB	0.406	0.166	.276	2.445	.018	0.072	0.740	.841	1.189
Peer_ASB	0.492	0.125	.453	3.946	.000	0.241	0.743	.810	1.235
Substance_Use	0.231	0.083	.298	2.794	.008	0.065	0.398	.937	1.067

a. Dependent Variable: AntiSocial_Beh

Significant predictor (p < .05)

What tolerance and VIF say about each predictor
Predictor	Tolerance	Already explained by the others (1 − tolerance)	VIF	SE of B inflated by
Parental ASB	.84	16%	1.19	1.09×
Peer ASB	.81	19%	1.24	1.11×
Substance use	.94	6%	1.07	1.03×

The last column is √VIF: how much wider the standard error of B is than it would be if the predictors were unrelated.

Step 9

APA 7th edition results table

Table 1

Multiple Regression Analysis Predicting Anti-Social Behaviour From Parental Anti-Social Behaviour, Peer Anti-Social Behaviour and Substance Use

Predictor	B	95% CI	SE B	β	t	p
Constant	−3.36	[−7.33, 0.62]	1.97		−1.70	.096
Parental anti-social behaviour	0.41	[0.07, 0.74]	0.17	.28*	2.45	.018
Peer anti-social behaviour	0.49	[0.24, 0.74]	0.12	.45***	3.95	<.001
Substance use	0.23	[0.06, 0.40]	0.08	.30**	2.79	.008

Note. N = 50. R² = .51, adjusted R² = .48, F(3, 46) = 15.86, p < .001. CI = confidence interval for B.
*p < .05. **p < .01. ***p < .001.

Step 10

Example write-up (APA 7th edition)

A multiple regression was carried out to examine whether parental anti-social behaviour, peer anti-social behaviour and substance use predicted young people's anti-social behaviour (N = 50). The normal P-P plot indicated that the residuals were approximately normally distributed, and no Cook's distance value exceeded 1 (maximum = .26), suggesting there were no influential cases. The Durbin-Watson statistic (1.72) indicated that the residuals were independent.

Anti-social behaviour was significantly positively correlated with parental anti-social behaviour, r(48) = .43, p = .001, peer anti-social behaviour, r(48) = .62, p < .001, and substance use, r(48) = .37, p = .004. Correlations between the predictors were weak to moderate (rs = −.06 to .37), and the collinearity statistics were acceptable (tolerance = .81 to .94; VIF = 1.07 to 1.24), indicating no collinearity.

The model was significant, F(3, 46) = 15.86, p < .001, R = .71, and explained 51% of the variance in anti-social behaviour (R² = .51, adjusted R² = .48). All three predictors were significant. Peer anti-social behaviour was the strongest predictor, B = 0.49, 95% CI [0.24, 0.74], β = .45, t(46) = 3.95, p < .001, followed by substance use, B = 0.23, 95% CI [0.06, 0.40], β = .30, t(46) = 2.79, p = .008, and parental anti-social behaviour, B = 0.41, 95% CI [0.07, 0.74], β = .28, t(46) = 2.45, p = .018. See Table 1.

Step 11

Glossary

Outcome / DV: The variable being predicted (anti-social behaviour).
Predictor / IV: A variable used to predict the outcome.
Residual: The difference between a person's actual score and the score the model predicts: the error.
Pearson's r: Strength and direction of a linear relationship, from −1 to +1.
R: Correlation between observed and predicted outcome scores.
R²: Proportion of variance in the outcome explained by the model (× 100 = %).
Adjusted R²: R² corrected for the number of predictors; an estimate for the population.
F-ratio: Variance explained by the model divided by unexplained variance. Tests the whole model.
B: Unstandardised coefficient: change in outcome per one-unit increase in the predictor.
SE B: Standard error of B: how precisely B is estimated.
β (Beta): Standardised coefficient. Use it to compare predictor strength.
t: B ÷ SE B. Tests whether a predictor is significant.
p-value: Probability of a result at least this extreme if there were no real effect. Below .05 = significant.
95% CI: Range likely to contain the true B. Crossing zero = non-significant.
Collinearity: When predictors are too highly correlated, making it hard to separate their effects. It blurs the individual coefficients, but not the model as a whole.
Tolerance: 1 − R² of a predictor regressed on the others: the share of its information that is unique. Close to 1 is good, below .2 is a concern, below .1 is serious.
VIF: Variance Inflation Factor = 1 ÷ Tolerance: how much the overlap inflates the standard error of B. Close to 1 is good, above 5 is worth mentioning, above 10 is a problem.
Durbin-Watson: Tests independence of residuals on a 0-to-4 scale. 1 to 3 is acceptable and 2 is ideal; below 1 suggests positive autocorrelation, above 3 negative.
Independence of residuals: Each person's error tells you nothing about anyone else's. Needed for the standard errors, t and p to be trustworthy.
Autocorrelation: Serial dependence: one residual predicts the next, so the errors are not independent.
P-P plot: Plots residuals against a normal distribution. Points on the line = normal.
Leverage (hat value): How far a case sits from the others on the predictors, i.e. how much pull it could have on the line. Average = (k + 1) ÷ N.
Mahalanobis D²: How unusual a case is on the predictors combined, compared with the rest of the sample. Above 7.815 (df = 3) is unusual at p = .05.
Cook's distance: How much the whole model would move if a case were deleted. Needs both a large residual and leverage. Above 1 = influential outlier; above 4 ÷ N is worth checking.