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Introduction to Regression Procedures

Comments on Interpreting Regression Statistics

In most applications, regression models are merely useful approximations. Reality is often so complicated that you cannot know what the true model is. You may have to choose a model more on the basis of what variables can be measured and what kinds of models can be estimated than on a rigorous theory that explains how the universe really works. However, even in cases where theory is lacking, a regression model may be an excellent predictor of the response if the model is carefully formulated from a large sample. The interpretation of statistics such as parameter estimates may nevertheless be highly problematical.

Statisticians usually use the word "prediction" in a technical sense. Prediction in this sense does not refer to "predicting the future" (statisticians call that forecasting) but rather to guessing the response from the values of the regressors in an observation taken under the same circumstances as the sample from which the regression equation was estimated. If you developed a regression model for predicting consumer preferences in 1958, it may not give very good predictions in 1988 no matter how well it did in 1958. If it is the future you want to predict, your model must include whatever relevant factors may change over time. If the process you are studying does in fact change over time, you must take observations at several, perhaps many, different times. Analysis of such data is the province of SAS/ETS procedures such as AUTOREG and STATESPACE. Refer to the SAS/ETS User's Guide for more information on these procedures.

The comments in the rest of this section are directed toward linear least-squares regression. Nonlinear regression and non-least-squares regression often introduce further complications.

For more detailed discussions of the interpretation of regression statistics, see Darlington (1968), Mosteller and Tukey (1977), Weisberg (1985), and Younger (1979).

Interpreting Parameter Estimates from a Controlled Experiment

Parameter estimates are easiest to interpret in a controlled experiment in which the regressors are manipulated independently of each other. In a well-designed experiment, such as a randomized factorial design with replications in each cell, you can use lack-of-fit tests and estimates of the standard error of prediction to determine whether the model describes the experimental process with adequate precision. If so, a regression coefficient estimates the amount by which the mean response changes when the regressor is changed by one unit while all the other regressors are unchanged. However, if the model involves interactions or polynomial terms, it may not be possible to interpret individual regression coefficients. For example, if the equation includes both linear and quadratic terms for a given variable, you cannot physically change the value of the linear term without also changing the value of the quadratic term. Sometimes it may be possible to recode the regressors, for example by using orthogonal polynomials, to make the interpretation easier.

If the nonstatistical aspects of the experiment are also treated with sufficient care (including such things as use of placebos and double blinds), then you can state conclusions in causal terms; that is, this change in a regressor causes that change in the response. Causality can never be inferred from statistical results alone or from an observational study.

If the model that you fit is not the true model, then the parameter estimates may depend strongly on the particular values of the regressors used in the experiment. For example, if the response is actually a quadratic function of a regressor but you fit a linear function, the estimated slope may be a large negative value if you use only small values of the regressor, a large positive value if you use only large values of the regressor, or near zero if you use both large and small regressor values. When you report the results of an experiment, it is important to include the values of the regressors. It is also important to avoid extrapolating the regression equation outside the range of regressors in the sample.

Interpreting Parameter Estimates from an Observational Study

In an observational study, parameter estimates can be interpreted as the expected difference in response of two observations that differ by one unit on the regressor in question and that have the same values for all other regressors. You cannot make inferences about "changes" in an observational study since you have not actually changed anything. It may not be possible even in principle to change one regressor independently of all the others. Neither can you draw conclusions about causality without experimental manipulation.

If you conduct an observational study and if you do not know the true form of the model, interpretation of parameter estimates becomes even more convoluted. A coefficient must then be interpreted as an average over the sampled population of expected differences in response of observations that differ by one unit on only one regressor. The considerations that are discussed under controlled experiments for which the true model is not known also apply.

Comparing Parameter Estimates

Two coefficients in the same model can be directly compared only if the regressors are measured in the same units. You can make any coefficient large or small just by changing the units. If you convert a regressor from feet to miles, the parameter estimate is multiplied by 5280.

Sometimes standardized regression coefficients are used to compare the effects of regressors measured in different units. Standardizing the variables effectively makes the standard deviation the unit of measurement. This makes sense only if the standard deviation is a meaningful quantity, which usually is the case only if the observations are sampled from a well-defined population. In a controlled experiment, the standard deviation of a regressor depends on the values of the regressor selected by the experimenter. Thus, you can make a standardized regression coefficient large by using a large range of values for the regressor.

In some applications you may be able to compare regression coefficients in terms of the practical range of variation of a regressor. Suppose that each independent variable in an industrial process can be set to values only within a certain range. You can rescale the variables so that the smallest possible value is zero and the largest possible value is one. Then the unit of measurement for each regressor is the maximum possible range of the regressor, and the parameter estimates are comparable in that sense. Another possibility is to scale the regressors in terms of the cost of setting a regressor to a particular value, so comparisons can be made in monetary terms.

Correlated Regressors

In an experiment, you can often select values for the regressors such that the regressors are orthogonal (not correlated with each other). Orthogonal designs have enormous advantages in interpretation. With orthogonal regressors, the parameter estimate for a given regressor does not depend on which other regressors are included in the model, although other statistics such as standard errors and p-values may change.

If the regressors are correlated, it becomes difficult to disentangle the effects of one regressor from another, and the parameter estimates may be highly dependent on which regressors are used in the model. Two correlated regressors may be nonsignificant when tested separately but highly significant when considered together. If two regressors have a correlation of 1.0, it is impossible to separate their effects.

It may be possible to recode correlated regressors to make interpretation easier. For example, if X and Y are highly correlated, they could be replaced in a linear regression by X+Y and X-Y without changing the fit of the model or statistics for other regressors.

Errors in the Regressors

If there is error in the measurements of the regressors, the parameter estimates must be interpreted with respect to the measured values of the regressors, not the true values. A regressor may be statistically nonsignificant when measured with error even though it would have been highly significant if measured accurately.

Probability Values (p-values)

Probability values (p-values) do not necessarily measure the importance of a regressor. An important regressor can have a large (nonsignificant) p-value if the sample is small, if the regressor is measured over a narrow range, if there are large measurement errors, or if another closely related regressor is included in the equation. An unimportant regressor can have a very small p-value in a large sample. Computing a confidence interval for a parameter estimate gives you more useful information than just looking at the p-value, but confidence intervals do not solve problems of measurement errors in the regressors or highly correlated regressors.

The p-values are always approximations. The assumptions required to compute exact p-values are never satisfied in practice.

Interpreting R2

R2 is usually defined as the proportion of variance of the response that is predictable from (that can be explained by) the regressor variables. It may be easier to interpret \sqrt{1-R^2},which is approximately the factor by which the standard error of prediction is reduced by the introduction of the regressor variables.

R2 is easiest to interpret when the observations, including the values of both the regressors and response, are randomly sampled from a well-defined population. Nonrandom sampling can greatly distort R2. For example, excessively large values of R2 can be obtained by omitting from the sample observations with regressor values near the mean.

In a controlled experiment, R2 depends on the values chosen for the regressors. A wide range of regressor values generally yields a larger R2 than a narrow range. In comparing the results of two experiments on the same variables but with different ranges for the regressors, you should look at the standard error of prediction (root mean square error) rather than R2.

Whether a given R2 value is considered to be large or small depends on the context of the particular study. A social scientist might consider an R2 of 0.30 to be large, while a physicist might consider 0.98 to be small.

You can always get an R2 arbitrarily close to 1.0 by including a large number of completely unrelated regressors in the equation. If the number of regressors is close to the sample size, R2 is very biased. In such cases, the adjusted R2 and related statistics discussed by Darlington (1968) are less misleading.

If you fit many different models and choose the model with the largest R2, all the statistics are biased and the p-values for the parameter estimates are not valid. Caution must be taken with the interpretation of R2 for models with no intercept term. As a general rule, no-intercept models should be fit only when theoretical justification exists and the data appear to fit a no-intercept framework. The R2 in those cases is measuring something different (refer to Kvalseth 1985).

Incorrect Data Values

All regression statistics can be seriously distorted by a single incorrect data value. A decimal point in the wrong place can completely change the parameter estimates, R2, and other statistics. It is important to check your data for outliers and influential observations. The diagnostics in PROC REG are particularly useful in this regard.

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