Fitting Linear Models
Simple Linear Regression
In socalled simple linear regression we observe the response y_{i} and one quantitative covariate x_{i} for the ith individual. The mean for the ith individual is
Here is how this model is fit in R.
The regression coefficients are reported in the table
Coefficients: Estimate Std. Error t value Pr(>t) (Intercept) 0.1908 1.7410 0.11 0.913 x 1.0221 0.0740 13.81 <2e16 ***The estimate of β_{1} is −0.1908 and the estimate of β_{2} is 1.0221.
The R function lm
(online
help) fits the linear model.
The expression y ~ x
specifies the model described above.
More about model formulas is given in the R documentation for model formulas, but you don't need to look at that. We'll cover all the formulas you need to know about in this web page.
For now it's enough to know that the formula specifies the
predictor x
and the response y
.
If these variables were instead named fred
and sally
,
respectively, the model formula would have to be sally ~ fred
and the whole model fitting statement would be
out < lm(sally ~ fred)
Note that the model formula doesn't mention regression coefficients explicitly.
There is one regression coefficient for each predictor in the formula
(that is, there is a β_{2} for the predictor x
)
and there is also a regression coefficient for the constant predictor
(that is, there is a β_{1} too) included by default.
The reason why the model fit is saved in a variable out
is because we generally want to do many things with it. In this example,
we use it twice. In other examples, we will use it more.
Quadratic Regression
It's called linear regression
because the means are linear functions
of the regression coefficients not because the means are linear functions
of the covariates — they may be arbitrary functions of the covariates.
In socalled quadratic regression we observe the response y_{i} and one quantitative covariate x_{i} for the ith individual. The mean for the ith individual is
Try 1
Here is how this model is fit in R to the data used in the preceding section.
Again the regression coefficients are in the Estimate
column
of the Coefficients:
table.
For now it's enough to know that the formula specifies the
predictors x
and x^2
and the response y
.
The plus sign (+
) is magical. It separates different predictors.
The I
function that wraps
the second predictor function.
It is necessary to write I(x^2)
instead of just x^2
because otherwise the hat (^
) would not be interpreted
correctly.
Generally, you have to wrap
all complicated expressions
for predictors. For example, you would write I(sin(x))
to use sin(x)
as a predictor.
Only simple variable names do not have to be wrapped
.
Note that the model formula doesn't mention regression coefficients explicitly.
There is one regression coefficient for each predictor in the formula,
that is, there is a β_{2} for the predictor x
and a β_{3} for the predictor I(x^2)
,
and there is also a regression coefficient for the constant predictor
(that is, there is a β_{1} too) included by default.
The curve
function that draws the sample regression curve
on the plot we will just treat as magic
because it is too complicated
to explain. It is, of course, documented in the
online help, but you don't need to look at that.
The predict
function is explained in the section on confidence intervals for the regression function below.
For comparison, we add (dotted line) the regression line from the other plot (simple linear regression).
Try 2
Here is another way to fit the same model used in the preceding section to the same data.
The estimated regression function (the mean values considered as a function of
the covariate) is the same as before, but the regression coefficients are
different. That is because the R function poly
(online
help) uses a different basis for the quadratic function. The means
have the form
where g_{1} is a linear function
and g_{2} is a quadratic function.
It doesn't matter what linear function and quadratic function are used.
The class of functions is still all quadratic functions of x
.
This is an example of regression
coefficients are meaningless.
Proof that Mean Vectors are the Same
Here we check what we already know: the means are the same even though the coefficients are different.
The mean vectors μ = M β are computed by the R function
predict
,
which is explained in the section on confidence intervals for the regression function below.
Try 3
Here is the same model fit to different data having a much stronger quadratic part.
Multiple Regression
In socalled multiple regression we observe the response vector y and two (or more) quantitative covariates x1 and x2. The mean for the ith individual is
The ith row of the model matrix is thus
Here's how R fits this model.
Note that we can't call both predictor variables the same name.
Here we call them x1
and x2
.
Now there is no simple plot that shows the regression function (means as a function of covariates), because with two covariates and one response, the graph of the function is threedimensional.
Quadratic Multiple Regression
For the same data as the preceding section, we can model means as a quadratic function of the covariates. The mean for the ith individual is
Try 1
Here's how R fits this model.
Try 2
Here's another way R can fit this model.
As in the section using the poly function above, the same model is being fit, because the family of regression functions is the same — all bivariate quadratic functions of covariates — and the vector subspace of all mean values is the same. But the regression coefficients are different because a different model matrix is used.
Hypothesis Tests
Tests about Regression Coefficients
Done by R
A hypothesis test about whether a regression coefficient is zero
is automatically done for each regression coefficient by
the R function summary
. These tests are not corrected
for multiple testing and hence must be used with extreme caution.
We'll use the example for simple linear regression and the example for quadratic regression which were done above as examples.
The form below redoes those examples, except for the plots, which we don't need.
The columns labeled t value
and Pr(>t)
of the Coefficients:
table give, respectively,

t value
, the test statistic for a test of the hypothesis that the specified regression coefficient (the one for that line of the table) is actually zero. 
Pr(<t)
the Pvalue for the twotailed test about that regression coefficient.
For example, if we have done simple linear regression, and want to test hypotheses
H_{1} : β_{2} ≠ 0
then the value of the test statistic reported by R is T = 13.81. Assuming the null hypothesis and assuming the standard regression assumptions (IID normal errors), this test statistic has a Student t distribution with n − 2 degrees of freedom. The corresponding Pvalue reported by R is P < 2 × 10^{− 16}. (twotailed). But you don't have to do a lookup in a table of the t distribution, R does it for you.
The interpretation of for this test is, of course, that P < 2 × 10^{− 16} means the null hypothesis is rejected and the true unknown population regression coefficient β_{2} is nonzero.
If you wanted a onetailed rather than a twotailed test, you would divide the given Pvalue by 2.
For another example, if we have done quadratic regression, and want to test hypotheses
H_{1} : β_{3} ≠ 0
then the value of the test statistic is T = −1.320 and the Pvalue is 0.195.
The interpretation of this Pvalue is, of course, that the null hypothesis is accepted and the true unknown population regression coefficient is zero. As usual, that doesn't prove the population regression coefficient is actually zero. It only says that the data at hand don't give any real evidence it is not zero.
Done by Hand
The by hand
here doesn't mean completely by hand. R does most of
the work but leaves a bit for you.
Since the calculations are very much like the calculations for confidence intervals, we will put this out of sequence, in the section on confidence intervals in the subsection on general tests about regression coefficients.
Summary
The quadratic regression is unnecessary for these data. It appears that the linear model with the regression function
fits the data well.
The linear regression does appear to be necessary for these data. It appears that the model with the constant regression function
does not fit the data well. So the the linear term in the model is necessary.
Tests about Correlation Coefficients
Regression coefficients in simple linear regression are intimately related to correlation coefficients. The simple relation
relates the two. Thus the slope of the population regression line is zero if and only if the correlation of X and Y is zero. And thus the test with null hypothesis β_{2} = 0 done above is also a test with null hypothesis ρ = 0.
Omnibus Tests
Also of interest in multiple regression is a test of whether there are any regression coefficients that are significantly nonzero except for the coefficient β_{1} that goes with the constant predictor and is usually not of interest. That is we want to do a test of
H_{1} : β_{i} ≠ 0, for some i ≠ 1
This test is often not of particular research
interest. It serves
as a straw man to knock down
. The null hypothesis is generally
thought to be false and is easily rejected with a reasonable amount of data.
But it is important to do the test anyway. When the null hypothesis cannot be rejected, this means the data are completely worthless. The model that has the constant regression function fits as well as the regression model.
Thus if this test fails to reject H_{0} the right thing to do is throw the data in the trash. They're worthless. No further analysis need be done.
The idea that their precious data, obtained at great cost in time, money, or effort, might actually be worthless upsets some people so much that they don't even want to think about doing this test. But they should. Drawing conclusions from worthless data makes you a fool, whether or not you are aware of your foolishness. If someone else reanalyzes the data, they'll find out.
R does this test in every regression printout. For quadratic regression example were done above part of the printout was
Coefficients:
Estimate Std. Error t value Pr(>t)
(Intercept) 2.883215 2.670473 1.080 0.287
x 1.406719 0.300391 4.683 3.74e05 ***
I(x^2) 0.009381 0.007105 1.320 0.195

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.351 on 37 degrees of freedom
Multiple Rsquared: 0.8414, Adjusted Rsquared: 0.8328
Fstatistic: 98.11 on 2 and 37 DF, pvalue: 1.613e15
The last line describes this omnibus
test.
The interpretation is that the null hypothesis is rejected (P = 1.6 × 10^{− 15}). So the regression is not completely worthless. Either β_{1} or β_{2} appears to be nonzero (or perhaps both).
Model Comparison Tests
In this section we consider testing a big model
versus a little model
. The null hypothesis is some regression
model, and the alternative hypothesis is some other regression model,
and the little model is a submodel of the big model
(the little model is obtained by setting some of the regression coefficients
of the big model to zero).
As an example, let us consider the
multiple regression example done above.
We use the model from that fit as the little model
.
And we consider a quadratic model as the big model
.
A quadratic model has three more terms. The regression function
for the big model is
We obtain the regression function for the little model by setting β_{4} = β_{5} = β_{6} = 0. So the little model is indeed a submodel of the big model (as the test requires).
To compare the fits of the two models we first do the fits,
which we save in the R objects out.little
and out.big
. Then we compare the models using
the anova
function
(online
help).
The printout of that function
is a socalled ANOVA (analysis of variance) table, which dates
back to the days of hand calculation and gives old timers a
warm fuzzy feeling. For our purposes we only need to get
the test statistic (F = 1.392) and Pvalue
(P = 0.2578) out of this table.
The interpretation of the test is that the null hypothesis is accepted. The data give no evidence of statistically significant departure from the little model. That doesn't prove the little model is actually correct, only that the data give no evidence it isn't.
You can't get the same effect by looking at the three Pvalues for the three regression coefficients separately. If you want to make this model comparison, you need to do this test.
The omnibus test
of the preceding section is the special case
of the test of this section where the little model has only the constant
predictor.
Confidence and Prediction Intervals
Confidence Intervals for Regression Coefficients
We again use the example for quadratic regression done above.
R doesn't do confidence intervals for regression coefficients for you. It does most of the work but leaves a bit left for you. Let's again look at the regression printout for that example.
Coefficients: Estimate Std. Error t value Pr(>t) (Intercept) 2.883215 2.670473 1.080 0.287 x 1.406719 0.300391 4.683 3.74e05 *** I(x^2) 0.009381 0.007105 1.320 0.195  Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 5.351 on 37 degrees of freedom Multiple Rsquared: 0.8414, Adjusted Rsquared: 0.8328 Fstatistic: 98.11 on 2 and 37 DF, pvalue: 1.613e15
The printout gives us three numbers (highlighted in yellow) needed to construct the confidence interval. The same three numbers are also needed for general tests about regression coefficients described in the following section.
 The first number is the estimate, also called the sample regression coefficient.
 The second number is the standard error of this estimate (the estimated standard deviation of the sampling distribution of the estimate).
 The third number is the degrees of freedom for the t pivotal quantity made from the first two numbers.
Thus one makes up a confidence interval in the usual way
where the critical value here comes from the t distribution with the degrees of freedom stated.
If we want a 95% confidence interval, the critical value for 37 degrees for freedom is given by
Rweb:> qt(0.975, 37) [1] 2.026192
and then the confidence interval for the parameter β_{3} for these data is given by
Rweb:> 0.009381 + c(1, 1) * qt(0.975, 7) * 0.007105 [1] 0.026181655 0.007419655
Of course, this whole calculation can be done by hand with a calculator and a table of the t distribution starting with the regression output. No further use of R is necessary after getting the regression output.
General Tests about Regression Coefficients
General test (with the hypothesized value under the null hypothesis some
number other than zero) are done by hand
in the same way as
the confidence intervals.
We use the same three numbers highlighted above.
Suppose we want to test
H_{1} : β_{3} < 0.25
The test statistic for a t test is
In this example that is
Rweb:> (0.009381  0.25) / 0.007105 [1] 36.50683
And we look up the Pvalue using the pt
function.
Rweb:> tstat < (0.009381  0.25) / 0.007105 Rweb:> pt(tstat, 37) [1] 6.359019e31
Highly statistically significant difference.
Confidence Intervals for the Regression Function
The R function predict
(online
help) makes confidence intervals for means and prediction intervals
for new data.
Mean Values for Old Data
Estimates of the (unknown, true) mean values for the observed
data are done by the predict
function using the defaults for
all arguments except the first.
We again use the data for the quadratic regression example.
Confidence Intervals for Means for Old Data
Confidence intervals for the (unknown, true) mean values for the observed
data are done by the predict
function using the optional
argument interval = "confidence"
.
Mean Values for New Data
Estimates of the (unknown, true) mean values for new
data are done by the predict
function using
optional argument newdata
, which should be a data frame
(online
help) that contains variables having the same names as the predictors
used in the model formula in the regression fit. This data frame
for the new data does not need to contain the response variable or unused
predictor variables.
Confidence Intervals for Means for New Data
Combining the methods of the two preceding sections, we get confidence intervals for the (unknown, true) mean values for new data.
Confidence Intervals for Means for New Data, Multiple Covariates
When there are multiple covariates, the data frame for the new data contains all covariates involved in the formula. We use the multiple regression example.
Prediction Intervals
Prediction intervals are also done by
the R function predict
.
Just use interval = "prediction"
instead of interval = "confidence"
.
Prediction Interval Plot
The middle curve is the estimated regression function. The upper and lower curves are the 95% prediction bounds.
To get confidence bounds for the regression function, change
type < "prediction"
to
type < "confidence"
.