Multiple linear regression

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This learning resource summarises the main teaching points about multiple linear regression (MLR), including key concepts, principles, assumptions, and how to conduct and interpret MLR analyses.

Prerequisites:

What is MLR?

Multiple linear regression (MLR) is a multivariate statistical technique for examining the linear correlations between two or more independent variables (IVs) and a single dependent variable (DV).
Research questions suitable for MLR can be of the form "To what extent do X1, X2, and X3 (IVs) predict Y (DV)?"
e.g., "To what extent does people's age and gender (IVs) predict their levels of blood cholesterol (DV)?"
MLR analyses can be visualised as path diagrams and/or venn diagrams

MLR studies the relation between two or more IVs and a single DV.
MLR is used to statistically 'distill' the relative contribution of two or more IVs on a single DV.
What other ways can you think of to explain the purpose of MLR?

Assumptions

View the accompanying screencast:

Level of measurement
1. IVs: Two or more continuous (interval or ratio) or dichotomous variables - it may be necessary to recode multichotomous categorical or ordinal IVs and non-normal interval or ratio IVs into dichotomous variables or a series of dummy variables)
2. DV: One continuous (interval or ratio) variable
Sample size (N; some rules of thumb):
1. Enough data is needed to provide reliable estimates of the correlations. Use at least 50 cases and at least 10 to 20 as many cases as there are IVs (as the number of IVs increases, more inferential tests are being conducted (if testing each predictor), therefore more data is needed), otherwise the estimates of the regression line are probably unstable and are unlikely to replicate if the study is repeated.
2. Green (2001) and Tabachnick and Fidell (2007) suggest:
  1. 50 + 8(k) for testing an overall regression model and
  2. 104 + k when testing individual predictors (where k is the number of IVs)
  3. These sample size suggestions are based on detecting a medium effect size (β >= .20), with critical α <= .05, with power of 80%.
    Study-specific power and sample size calculations should be conducted (e.g., http://www.danielsoper.com/statcalc3/calc.aspx?id=1; note that this calculator uses f² as the effect size - see the formula link for how to convert R² to to f²).
Normality
1. Check the univariate descriptive statistics (M, SD, skewness and kurtosis)
2. Check the histograms with a normal curve imposed
3. Estimates of correlations will be more reliable and stable when the variables are normally distributed
Linearity
1. Are the bivariate relationships linear?
2. Check scatterplots and correlations between the DV (Y) and each of the IVs (Xs)
3. Check for influence of bivariate outliers
Homoscedasticity
1. Are the bivariate distributions reasonably evenly spread about the line of best fit?
2. Check scatterplots between Y and each of Xs and/or check scatterplot of the residuals (ZRESID) and predicted values (ZPRED))
Multicollinearity
1. Is there multicollinearity between the IVs? Predictors should not be overly correlated with one another. Ways to check:
  1. Examine bivariate correlations and scatterplots between each of the IVs (i.e., are the predictors overly correlated e.g., above .7?).
  2. Check the collinearity statistics in the coefficients table:
    1. The Variance Inflation Factor (VIF) should be low (< ~3-10) and/or
    2. Tolerance should be high (> .1 to .3) (Note that TOL=1/VIF so only one needs to be used).
Multivariate outliers (MVOs)
1. Check whether there are influential MVOs using Mahalanobis' Distance (MD) and/or Cook’s D (CD).
2. SPSS: Linear Regression - Save - Mahalanobis (can also include Cook's D)
  1. After execution, new variables called mah_1 (and coo_1) will be added to the data file.
  2. In the output, check the Residuals Statistics table for the maximum MD and CD.
  3. The maximum MD should not exceed the critical chi-square value with degrees of freedom (df) equal to number of predictors, with critical alpha =.001. CD should not be greater than 1.
3. If outliers are detected, go to the data file, sort the data in descending order by mah_1, and check the cases with mah_1 distances above the critical value (these cases have an unusual combination of responses for the variables in the analysis). Consider removing these cases and re-running the MLR. If the results are very similar (e.g., similar R² and conclusions for each of the predictors, then it is best to use the original results, i.e., including the multivariate outliers. If the results are different when the MVOs are not included, then these cases probably have had undue influence and it is best to report the results without these cases.
Normality of residuals
1. Residuals are more likely to be normally distributed if each of the variables normally distributed
2. Check histograms of all variables in an analysis
3. Normally distributed variables will enhance the MLR solution

References

Allen & Bennett 13.3.2.1 Assumptions (pp. 178-179)
Francis 5.1.4 Practical Issues and Assumptions (pp. 126-128)
Green, S. B. (1991). How many subjects does it take to do a regression analysis?. Multivariate Behavioral Research, 26, 499-510.
Knofczynski, G. T., & Mundfrom, D. (2008). Sample sizes when using multiple linear regression for prediction. Educational and Psychological Measurement, 68, 431-442.
Wilson Van Voorhis, C. R. & Morgan, B. L. (2007). Understanding power and rules of thumb for determining sample sizes. Tutorials in Quantitative Methods for Psychology, 3(2), 43-50.

Types

There are several types of MLR, including:

Type	Characteristics
Direct (or Standard)	All IVs are entered simultaneously
Hierarchical	IVs are entered in steps, i.e., some before others Interpret: R² change, F change
Forward	The software enters IVs one by one until there are no more significant IVs to be entered
Backward	The software removes IVs one to one until there are no more non-significant IVs to removed
Stepwise	A combination of Forward and Backward MLR

Results

MLR analyses produce several diagnostic and outcome statistics which are summarised below and are important to understand.
Make sure that you can learn how to find and interpret these statistics from statistical software output.

Correlations

Examine the linear correlations between (usually as a correlation matrix, but also view the scatterplots):

IVs
each IV and the DV

Effect sizes

R

(Big) R is the multiple correlation coefficient for the relationship between the predictor and outcome variables.
Interpretation is similar to that for little r (the linear correlation between two variables), however R can only range from 0 to 1, with 0 indicating no relationship and 1 a perfect relationship. Large values of R indicate more variance explained in the DV.
R can be squared and interpreted as for r², with a rough rule of thumb being .1 (small), .3 (medium), and .5 (large). These R² values would indicate 10%, 30%, and 50% of the variance in the DV explained respectively.
When generalising findings to the population, the R² for a sample tends to overestimate the R² of the population. Thus, adjusted R² is recommended when generalising from a sample, and this value will be adjusted downward based on the sample size; the smaller the sample size, the greater the reduction.
The statistical significance of R can be examined using an F test and its corresponding p level.
Reporting example: R^{2^{= .32, F(6, 217) = 19.50, p = .001
"6, 217" refers to the degrees of freedom - for more information, see about half-down this page}}

Cohen's ƒ²

Cohen's ƒ² is based on the R² and is an alternate indicator of effect size for MLR.

Coefficients

An MLR analysis produces several useful statistics about each of the predictors. These regression coefficients are usually presented in a Results table which may include:

B (unstandardised) - used for building a prediction equation
β (standardised) - indicates the relative strength of the predictors on a scale ranging from -1 to 1.
Zero-order correlation (r) - the correlation between a predictor and the outcome variable
Partial correlations (pr) - indicate the unique correlations between each IV and the DV (labelled "partial" in SPSS output)
Semi-part correlations (sr) - similar to partial correlations (labelled "part" in SPSS output); squaring this value provides the percentage of variance in the DV uniquely explained by each IV (sr²)
t, p - indicate the statistical significance of each IV
Confidence intervals - indicate the probably range of population values for the βs

Equation

A prediction equation can be derived from the regression coefficients in a MLR analysis.
The equation is of the form

$\hat{Y} = bx + a$ (for predicted values) or
$Y = bx + a + e$ (for observed values)

Residuals

A residual is the difference between the actual value of a DV and its predicted value. Each case will have a residual for each MLR analysis. Three key assumptions can be tested using plots of residuals:

Linearity: IVs are linearly related to DV
Normality of residuals
Equal variances (Homoscedasticity)

Power

Post-hoc statistical power calculator for MLR

Advanced concepts

Partial correlations
Use of hierarchical regression to partial out or remove the effect of 'control' variables
Interactions between IVs
Moderation and mediation

Writing up

When writing up the results of an MLR, consider describing:

Assumptions: How were they tested? To what extent were the assumptions met?
Correlations: What are they? Consider correlations between the IVs and the DV separately to the correlations between the IVs.
Regression coefficients: Report a table and interpret
Causality: Be aware of the limitations of the analysis - it may be consistent with a causal relationship, but it is unlikely to prove causality

FAQ

What if there are univariate outliers?

Basically, explore and consider what the implications might be - do these "outliers" impact on the assumptions? A lot depends on how "outliers" are defined. It is probably better to consider distributions in terms of the shape of the histogram and skewness and kurtosi, and whether these values are unduely impacting on the estimates of linear relations between variables. In other words, what are the implications? Ultimately, the researcher needs to decide whether the outliers are so severe that they are unduely influencing results of analyses or whether they are relatively benign. If unsure, explore, test, try the analyses with and without these values etc. If still unsure, be conservative and remove the data points or recode the data.

External links

Correlation and simple least squares regression (Zady, 2000)
Multiple regression (Statsoft)
Multiple regression assumptions (ERIC Digest)

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