Maximum number of independent variables that can be entered into a multiple regression equation

What is the limit to the number of independent variables one may enter in a multiple regression equation? I have 10 predictors that I would like to examine in terms of their relative contribution to the outcome variable. Should I use a bonferroni correction to adjust for multiple analyses?

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You need to think about what you mean by a "limit". There are limits, such as when you have more predictors than cases, you run into issues in parameter estimation (see the little R simulation at the bottom of this answer).

However, I imagine you are talking more about soft limits related to statistical power and good statistical practice. In this case the language of "limits" is not really appropriate. Rather, bigger sample sizes tend to make it more reasonable to have more predictors and the threshold of how many predictors is reasonable arguably falls on a continuum of reasonableness. You may find the discussion of rules of thumb for sample size in multiple regression relevant, as many such rules of thumb make reference to the number of predictors.

A few points

• If you are concerned more with overall prediction than with statistical significance of individual predictors, then it is probably reasonable to include more predictors than if you are concerned with statistical significance of individual predictors.
• If you are concerned more with testing a specific statistical model that relates to your research question (e.g., as is common in many social science applications), presumably you have reasons for including particular predictors. However, you may also have opportunities to be selective in which predictors you include (e.g., if you have multiple variables that measure a similar construct, you might only include one of them). When doing theory based model testing, there are a lot of choices, and the decision about which predictors to include involves close connection between your theory and research question.
• I don't often see researchers using bonferroni corrections being applied to significance tests of regression coefficients. One reasonable reason for this might be that researchers are more interested in appraising the overall properties of the model.
• If you are interested in assessing relative importance of predictors, I find it useful to examine both the bivariate relationship between the predictor and the outcome, as well as the relationship between the predictor and outcome controlling for other predictors. If you include many predictors, it is often more likely that you include predictors that are highly intercorrelated. In such cases, interpretation of both the bivariate and model based importance indices can be useful, as a variable important in a bivariate sense might be hidden in a model by other correlated predictors (I elaborate more on this here with links).

A little R simulation

I wrote this little simulation to highlight the relationship between sample size and parameter estimation in multiple regression.

set.seed(1)

fitmodel <- function(n, k) {
# n: sample size
# k: number of predictors
# return linear model fit for given sample size and k predictors
x <- data.frame(matrix( rnorm(n*k), nrow=n))
names(x) <- paste("x", seq(k), sep="")
x$y <- rnorm(n) lm(y~., data=x) }  The fitmodel function takes two arguments n for the sample size and k for the number of predictors. I am not counting the constant as a predictor, but it is estimated. I then generates random data and fits a regression model predicting a y variable from k predictor variables and returns the fit. Given that you mentioned in your question that you were interested in whether 10 predictors is too much, the following function calls show what happens when the sample size is 9, 10, 11, and 12 respectively. I.e., sample size is one less than the number of predictors to two more than the number of predictors summary(fitmodel(n=9, k=10)) summary(fitmodel(n=10, k=10)) summary(fitmodel(n=11, k=10)) summary(fitmodel(n=12, k=10))  > summary(fitmodel(n=9, k=10)) Call: lm(formula = y ~ ., data = x) Residuals: ALL 9 residuals are 0: no residual degrees of freedom! Coefficients: (2 not defined because of singularities) Estimate Std. Error t value Pr(>|t|) (Intercept) -0.31455 NA NA NA x1 0.34139 NA NA NA x2 -0.45924 NA NA NA x3 0.42474 NA NA NA x4 -0.87727 NA NA NA x5 -0.07884 NA NA NA x6 -0.03900 NA NA NA x7 1.08482 NA NA NA x8 0.62890 NA NA NA x9 NA NA NA NA x10 NA NA NA NA Residual standard error: NaN on 0 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: NaN F-statistic: NaN on 8 and 0 DF, p-value: NA  Sample size is one less than the number of predictors. It is only possible to estimate 9 parameters, one of which is the constant. > summary(fitmodel(n=10, k=10)) Call: lm(formula = y ~ ., data = x) Residuals: ALL 10 residuals are 0: no residual degrees of freedom! Coefficients: (1 not defined because of singularities) Estimate Std. Error t value Pr(>|t|) (Intercept) 0.1724 NA NA NA x1 -0.3615 NA NA NA x2 -0.4670 NA NA NA x3 -0.6883 NA NA NA x4 -0.1744 NA NA NA x5 -1.0331 NA NA NA x6 0.3886 NA NA NA x7 -0.9886 NA NA NA x8 0.2778 NA NA NA x9 0.4616 NA NA NA x10 NA NA NA NA Residual standard error: NaN on 0 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: NaN F-statistic: NaN on 9 and 0 DF, p-value: NA  Sample size is the same as the number of predictors. It is only possible to estimate 10 parameters, one of which is the constant. > summary(fitmodel(n=11, k=10)) Call: lm(formula = y ~ ., data = x) Residuals: ALL 11 residuals are 0: no residual degrees of freedom! Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.9638 NA NA NA x1 -0.8393 NA NA NA x2 -1.5061 NA NA NA x3 -0.4917 NA NA NA x4 0.3251 NA NA NA x5 4.4212 NA NA NA x6 0.7614 NA NA NA x7 -0.4195 NA NA NA x8 0.2142 NA NA NA x9 -0.9264 NA NA NA x10 -1.2286 NA NA NA Residual standard error: NaN on 0 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: NaN F-statistic: NaN on 10 and 0 DF, p-value: NA  Sample size is one more than the number of predictors. All parameters are estimated including the constant. > summary(fitmodel(n=12, k=10)) Call: lm(formula = y ~ ., data = x) Residuals: 1 2 3 4 5 6 7 8 9 10 11 0.036530 -0.042154 -0.009044 -0.117590 0.171923 -0.007976 0.050542 -0.011462 0.010270 0.000914 -0.083533 12 0.001581 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.14680 0.11180 1.313 0.4144 x1 0.02498 0.09832 0.254 0.8416 x2 1.01950 0.13602 7.495 0.0844 . x3 -1.76290 0.26094 -6.756 0.0936 . x4 0.44832 0.16283 2.753 0.2218 x5 -0.76818 0.15651 -4.908 0.1280 x6 -0.33209 0.18554 -1.790 0.3244 x7 1.62276 0.21562 7.526 0.0841 . x8 -0.47561 0.18468 -2.575 0.2358 x9 1.70578 0.31547 5.407 0.1164 x10 3.25415 0.46447 7.006 0.0903 . --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.2375 on 1 degrees of freedom Multiple R-squared: 0.995, Adjusted R-squared: 0.9452 F-statistic: 19.96 on 10 and 1 DF, p-value: 0.1726  Sample size is two more than the number of predictors, and it is finally possible to estimate the fit of the overall model. - I often look at this from the standpoint of whether a model fitted with a certain number of parameters is likely to yield predictions out-of-sample that are as accurate as predictions made on the original model development sample. Calibration curves, mean squared errors of X*Beta, and indexes of predictive discrimination are some of the measures typically used. This is where some of the rules of thumb come from, such as the 15:1 rule (an effective sample size of 15 per parameter examined or estimated). Regarding multiplicity, a perfect adjustment for multiplicity, assuming the model holds and distributional assumptions are met, is the global test that all betas (other than the intercept) are zero. This is typically tested using a likelihood ratio or an F test. There are two overall approaches to model development that tend to work well. (1) Have an adequate sample size and fit the entire pre-specified model, and (2) used penalized maximum likelihood estimation to allow only as many effective degrees of freedom in the the regression as the current sample size will support. [Stepwise variable selection without penalization should play no role, as this is known not to work.] - Are those rules of thumb based on assumptions about the size of the true regression coefficients and the size of the error variance? Would I be right in thinking that if the error variance was known to be very small, a much smaller ratio of data points to parameters would be acceptable? – mark999 Jul 11 '11 at 1:58 That is an excellent point which I neglected to mention. The 15:1 rule is for the types of signal:noise ratios seen in biomedical and social sciences. When you have a low residual variance, you can estimate many more parameters accurately. – Frank Harrell Jul 11 '11 at 2:09 I would rephrase the question as follows: I have$n$observations, and$p$candidate predictors. Assume that the true model is a linear combination of$m$variables among the$p$candidate predictors. Is there an upper bound to$m$(your limit), such that I can still identify this model? Intuitively, if$m$is too large compared to$n$, or it is large compared to$p$, it may be hard to identify the correct model. In other terms: is there a limit to model selection? To this question, Candes and Plan give an affirmative answer in their paper "Near-ideal model selection by$\ell_1$minimization":$m \le K p \sigma_1/\log(p)$, where$\sigma_1$is the largest singular value of the matrix of predictors$X$. This is a deep result, and although it relies on several technical conditions, it links the number of observations (through$\sigma_1$) and of$p$to the number of predictors we can hope to estimate. - In principle, there is no limit per se to how many predictors you can have. You can estimate 2 billion "betas" in principle. But what happens in practice is that without sufficient data, or sufficient prior information, it will not prove a very fruitful exercise. No particular parameters will be determined very well, and you will not learn much from the analysis. Now if you don't have a lot of prior information about your model (model structure, parameter values, noise, etc.) then you will need the data to provide this information. This is usually the most common situation, which makes sense, because you usually need a pretty good reason to collect data (and spend$) about something you already know pretty well. If this is your situation, then a reasonable limit is to have a large number of observations per parameter. You have 12 parameters (10 slope betas, 1 intercept, and a noise parameter), so anything over 100 observations should be able to determine your parameters well enough to be able to make some conclusions.

But there is no "hard and fast" rules. With only 10 predictors you should have no trouble with computation time (get a better computer if you do). It mainly means just doing more work, because you have 11 dimensions of data to absorb - making it difficult to visualise the data. The basic principles from regression with only 1 dependent variable aren't really that different.

The problem with bonferroni correction is that for it to be a reasonable way to adjust your significance level without sacrificing too much power, you require the hypothesis that you are correcting for to be independent (i.e. learning that one hypothesis is true tells you nothing about whether another hypothesis is true). This is not true for the standard "t-test" in multiple regression for a co-efficient being zero, for example. The test statistic depends on what else in the model - which is a roundabout way of saying the hypothesis are dependent. Or, a more frequentist way of saying this is that the sampling distribution of the t-value conditional on the ith predictor being zero depends on what other parameters are zero. So using the bonferroni correction here may be actually be giving you a lower "overall" significance level than what you think.

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