When multiple variables are associated with a response, the interpretation of a prediction equation is seldom simple.
Last month we explored how to model a simple relationship between two variables, such as the dependence of weight on height^{1}. In the more realistic scenario of dependence on several variables, we can use multiple linear regression (MLR). Although MLR is similar to linear regression, the interpretation of MLR correlation coefficients is confounded by the way in which the predictor variables relate to one another.
In simple linear regression^{1}, we model how the mean of variable Y depends linearly on the value of a predictor variable X; this relationship is expressed as the conditional expectation E(YX) = β_{0} + β_{1}X. For more than one predictor variable X_{1}, . . ., X_{p}, this becomes β_{0} + Σβ_{j}X_{j}. As for simple linear regression, one can use the leastsquares estimator (LSE) to determine estimates b_{j} of the β_{j} regression parameters by minimizing the residual sum of squares, SSE = Σ(y_{i} − ŷ_{i})^{2}, where ŷ_{i} = b_{0} + Σ_{j}b_{j}xij. When we use the regression sum of squares, SSR = Σ(ŷ_{i} − Y^{−})^{2}, the ratio R^{2} = SSR/(SSR + SSE) is the amount of variation explained by the regression model and in multiple regression is called the coefficient of determination.
The slope β_{j} is the change in Y if predictor j is changed by one unit and others are held constant. When normality and independence assumptions are fulfilled, we can test whether any (or all) of the slopes are zero using a ttest (or regression Ftest). Although the interpretation of β_{j} seems to be identical to its interpretation in the simple linear regression model, the innocuous phrase “and others are held constant” turns out to have profound implications.
To illustrate MLR—and some of its perils—here we simulate predicting the weight (W, in kilograms) of adult males from their height (H, in centimeters) and their maximum jump height (J, in centimeters). We use a model similar to that presented in our previous column^{1}, but we now include the effect of J as E(WH,J) = β_{H}H + β_{J}J + β_{0} + ε, with β_{H} = 0.7, β_{J} = −0.08, β_{0} = −46.5 and normally distributed noise ε with zero mean and σ = 1 (Table 1). We set β_{J} negative because we expect a negative correlation between W and J when height is held constant (i.e., among men of the same height, lighter men will tend to jump higher). For this example we simulated a sample of size n = 40 with H and J normally distributed with means of 165 cm (σ = 3) and 50 cm (σ = 12.5), respectively.
Although the statistical theory for MLR seems similar to that for simple linear regression, the interpretation of the results is much more complex. Problems in interpretation arise entirely as a result of the sample correlation^{2} among the predictors. We do, in fact, expect a positive correlation between H and J—tall men will tend to jump higher than short ones. To illustrate how this correlation can affect the results, we generated values using the model for weight with samples of J and H with different amounts of correlation.
Let's look first at the regression coefficients estimated when the predictors are uncorrelated, r(H,J) = 0, as evidenced by the zero slope in association between H and J (Fig. 1a). Here r is the Pearson correlation coefficient^{2}. If we ignore the effect of J and regress W on H, we find Ŵ = 0.71H − 51.7 (R^{2} = 0.66) (Table 1 and Fig. 1b). Ignoring H, we find Ŵ = −0.088J + 69.3 (R^{2} = 0.19). If both predictors are fitted in the regression, we obtain Ŵ = 0.71H − 0.088J − 47.3 (R^{2} = 0.85). This regression fit is a plane in three dimensions (H, J, W) and is not shown in Figure 1. In all three cases, the results of the Ftest for zero slopes show high significance (P ≤ 0.005).
When the sample correlations of the predictors are exactly zero, the regression slopes (b_{H} and b_{J}) for the “one predictor at a time” regressions and the multiple regression are identical, and the simple regression R^{2} sums to multiple regression R^{2} (0.66 + 0.19 = 0.85; Fig. 2). The intercept changes when we add a predictor with a nonzero mean to satisfy the constraint that the leastsquares regression line goes through the sample means, which is always true when the regression model includes an intercept.
Balanced factorial experiments show a sample correlation of zero among the predictors when their levels have been fixed. For example, we might fix three heights and three jump heights and select two men representative of each combination, for a total of 18 subjects to be weighed. But if we select the samples and then measure the predictors and response, the predictors are unlikely to have zero correlation.
When we simulate highly correlated predictors r(H,J) = 0.9 (Fig. 1c), we find that the regression parameters change depending on whether we use one or both predictors (Table 1 and Fig. 1d). If we consider only the effect of H, the coefficient β_{H} = 0.7 is inaccurately estimated as b_{H} = 0.44. If we include only J, we estimate β_{J} = −0.08 inaccurately, and even with the wrong sign (b_{J} = 0.097). When we use both predictors, the estimates are quite close to the actual coefficients (b_{H} = 0.63, b_{J} = −0.056).
In fact, as the correlation between predictors r(H,J) changes, the estimates of the slopes (b_{H}, b_{J}) and intercept (b_{0}) vary greatly when only one predictor is fitted. We show the effects of this variation for all values of predictor correlation (both positive and negative) across 250 trials at each value (Fig. 2). We include negative correlation because although J and H are likely to be positively correlated, other scenarios might use negatively correlated predictors (e.g., lung capacity and smoking habits). For example, if we include only H in the regression and ignore the effect of J, b_{H} steadily decreases from about 1 to 0.35 as r(H,J) increases. Why is this? For a given height, larger values of J (an indicator of fitness) are associated with lower weight. If J and H are negatively correlated, as J increases, H decreases, and both changes result in a lower value of W. Conversely, as J decreases, H increases, and thus W increases. If we use only H as a predictor, J is lurking in the background, depressing W at low values of H and enhancing W at high levels of H, so that the effect of H is overestimated (b_{H} increases). The opposite effect occurs when J and H are positively correlated. A similar effect occurs for b_{J}, which increases in magnitude (becomes more negative) when J and H are negatively correlated. Supplementary Figure 1 shows the effect of correlation when both regression coefficients are positive.
When both predictors are fitted (Fig. 2), the regression coefficient estimates (b_{H}, b_{J}, b_{0}) are centered at the actual coefficients (β_{H}, β_{J}, β_{0}) with the correct sign and magnitude regardless of the correlation of the predictors. However, the standard error in the estimates steadily increases as the absolute value of the predictor correlation increases.
Neglecting important predictors has implications not only for R^{2}, which is a measure of the predictive power of the regression, but also for interpretation of the regression coefficients. Unconsidered variables that may have a strong effect on the estimated regression coefficients are sometimes called 'lurking variables'. For example, muscle mass might be a lurking variable with a causal effect on both body weight and jump height. The results and interpretation of the regression will also change if other predictors are added.
Given that missing predictors can affect the regression, should we try to include as many predictors as possible? No, for three reasons. First, any correlation among predictors will increase the standard error of the estimated regression coefficients. Second, having more slope parameters in our model will reduce interpretability and cause problems with multiple testing. Third, the model may suffer from overfitting. As the number of predictors approaches the sample size, we begin fitting the model to the noise. As a result, we may seem to have a very good fit to the data but still make poor predictions.
MLR is powerful for incorporating many predictors and for estimating the effects of a predictor on the response in the presence of other covariates. However, the estimated regression coefficients depend on the predictors in the model, and they can be quite variable when the predictors are correlated. Accurate prediction of the response is not an indication that regression slopes reflect the true relationship between the predictors and the response.
References
Altman, N. & Krzywinski, M. Nat. Methods 12, 999–1000 (2015).
Altman, N. & Krzywinski, M. Nat. Methods 12, 899–900 (2015).
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Integrated supplementary information
Supplementary Figure 1 Regression coefficients and R^{2}
The significance and value of regression coefficients and R^{2} for a model with both regression coefficients positive, E(WH,J) = 0.7H + 0.08J  46.5 + ε. The format of the figure is the same as that of Figure 2.
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Supplementary Figure 1
Regression coefficients and R^{2} (PDF 299 kb)
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Krzywinski, M., Altman, N. Multiple linear regression. Nat Methods 12, 1103–1104 (2015). https://doi.org/10.1038/nmeth.3665
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DOI: https://doi.org/10.1038/nmeth.3665
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