lm is used to fit linear models.
It can be used to carry out regression,
single stratum analysis of variance and
analysis of covariance (although aov may provide a more
convenient interface for these).
The version distributed through the package mixlm
extends the capabilities with balanced mixture
models and lmer interfacing. Random effects
are indicated by wrapping their formula entries in r().
Also, effect level names are kept in printing.
lm(formula, data, subset, weights, na.action,
method = "qr", model = TRUE, x = TRUE, y = TRUE, qr = TRUE,
singular.ok = TRUE, contrasts = "contr.sum", offset,
unrestricted = TRUE, REML = NULL, ...)lm returns an object of class
c("lmm","lm") or for
multiple responses of class c("mlm", "lm").
The functions summary and anova are used to
obtain and print a summary and analysis of variance table of the
results. The generic accessor functions coefficients,
effects, fitted.values and residuals extract
various useful features of the value returned by lm.
An object of class "lm" is a list containing at least the
following components:
a named vector of coefficients
the residuals, that is response minus fitted values.
the fitted mean values.
the numeric rank of the fitted linear model.
(only for weighted fits) the specified weights.
the residual degrees of freedom.
the matched call.
the terms object used.
(only where relevant) the contrasts used.
(only where relevant) a record of the levels of the factors used in fitting.
the offset used (missing if none were used).
if requested, the response used.
if requested, the model matrix used.
if requested (the default), the model frame used.
(where relevant) information returned by
model.frame on the special handling of NAs.
In addition, non-null fits will have components assign,
effects and (unless not requested) qr relating to the linear
fit, for use by extractor functions such as summary and
And models containing random effect will contain
random having additional information about
the model.
an object of class "formula" (or one that
can be coerced to that class): a symbolic description of the
model to be fitted. The details of model specification are given
under ‘Details’.
an optional data frame, list or environment (or object
coercible by as.data.frame to a data frame) containing
the variables in the model. If not found in data, the
variables are taken from environment(formula),
typically the environment from which lm is called.
an optional vector specifying a subset of observations to be used in the fitting process.
an optional vector of weights to be used in the fitting
process. Should be NULL or a numeric vector.
If non-NULL, weighted least squares is used with weights
weights (that is, minimizing sum(w*e^2)); otherwise
ordinary least squares is used. See also ‘Details’,
a function which indicates what should happen
when the data contain NAs. The default is set by
the na.action setting of options, and is
na.fail if that is unset. The ‘factory-fresh’
default is na.omit. Another possible value is
NULL, no action. Value na.exclude can be useful.
the method to be used; for fitting, currently only
method = "qr" is supported; method = "model.frame" returns
the model frame (the same as with model = TRUE, see below).
logicals. If TRUE the corresponding
components of the fit (the model frame, the model matrix, the
response, the QR decomposition) are returned.
logical. If FALSE (the default in S but
not in R) a singular fit is an error.
character indicating which coding should be applied
to all factors or an optional list. See the contrasts.arg
of model.matrix.default. Defaults to "contr.sum".
See Details for more information.
this can be used to specify an a priori known
component to be included in the linear predictor during fitting.
This should be NULL or a numeric vector of length equal to
the number of cases. One or more offset terms can be
included in the formula instead or as well, and if more than one are
specified their sum is used. See model.offset.
additional argument for switching between unrestricted and restricted models if including random variables.
is used to invoke restricted maximum likelihood (TRUE) or maximum likelihood (FALSE) estimation instead of least squares.
additional arguments to be passed to the low level regression fitting functions (see below).
Considerable care is needed when using lm with time series.
Unless na.action = NULL, the time series attributes are
stripped from the variables before the regression is done. (This is
necessary as omitting NAs would invalidate the time series
attributes, and if NAs are omitted in the middle of the series
the result would no longer be a regular time series.)
Even if the time series attributes are retained, they are not used to
line up series, so that the time shift of a lagged or differenced
regressor would be ignored. It is good practice to prepare a
data argument by ts.intersect(..., dframe = TRUE),
then apply a suitable na.action to that data frame and call
lm with na.action = NULL so that residuals and fitted
values are time series.
The design was inspired by the S function of the same name described in Chambers (1992). The implementation of model formula by Ross Ihaka was based on Wilkinson & Rogers (1973). Mixed model extensions by Kristian Hovde Liland.
Models for lm are specified symbolically. A typical model has
the form response ~ terms where response is the (numeric)
response vector and terms is a series of terms which specifies a
linear predictor for response. A terms specification of the form
first + second indicates all the terms in first together
with all the terms in second with duplicates removed. A
specification of the form first:second indicates the set of
terms obtained by taking the interactions of all terms in first
with all terms in second. The specification first*second
indicates the cross of first and second. This is
the same as first + second + first:second.
If the formula includes an offset, this is evaluated and
subtracted from the response.
If response is a matrix a linear model is fitted separately by
least-squares to each column of the matrix.
See model.matrix for some further details. The terms in
the formula will be re-ordered so that main effects come first,
followed by the interactions, all second-order, all third-order and so
on: to avoid this pass a terms object as the formula (see
aov and demo(glm.vr) for an example).
A formula has an implied intercept term. To remove this use either
y ~ x - 1 or y ~ 0 + x. See formula for
more details of allowed formulae.
Non-NULL weights can be used to indicate that different
observations have different variances (with the values in
weights being inversely proportional to the variances); or
equivalently, when the elements of weights are positive
integers \(w_i\), that each response \(y_i\) is the mean of
\(w_i\) unit-weight observations (including the case that there are
\(w_i\) observations equal to \(y_i\) and the data have been
summarized).
lm calls the lower level functions lm.fit, etc,
see below, for the actual numerical computations. For programming
only, you may consider doing likewise.
All of weights, subset and offset are evaluated
in the same way as variables in formula, that is first in
data and then in the environment of formula.
The contrasts argument is applied when the lm model is
fitted. In additional to the standard contrasts from the stats
package, one can choose weighted coding: contr.weighted which
balances unbalanced data through factor weighting. Different from the
original version of lm, a single contrast can be indicated
to automatically be applied to all factors.
Chambers, J. M. (1992) Linear models. Chapter 4 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.
Wilkinson, G. N. and Rogers, C. E. (1973) Symbolic descriptions of factorial models for analysis of variance. Applied Statistics, 22, 392--9.
summary.lmm for summaries and anova.lmm for
the ANOVA table; aov for a different interface.
The generic functions coef, effects,
residuals, fitted, vcov.
predict.lm (via predict) for prediction,
including confidence and prediction intervals;
confint for confidence intervals of parameters.
lm.influence for regression diagnostics, and
glm for generalized linear models.
The underlying low level functions,
lm.fit for plain, and lm.wfit for weighted
regression fitting.
More lm() examples are available e.g., in
anscombe, attitude, freeny,
LifeCycleSavings, longley,
stackloss, swiss.
biglm in package biglm for an alternative
way to fit linear models to large datasets (especially those with many
cases).
print.AnovaMix, AnovaMix, Anova
require(graphics)
## Annette Dobson (1990) "An Introduction to Generalized Linear Models".
## Page 9: Plant Weight Data.
ctl <- c(4.17,5.58,5.18,6.11,4.50,4.61,5.17,4.53,5.33,5.14)
trt <- c(4.81,4.17,4.41,3.59,5.87,3.83,6.03,4.89,4.32,4.69)
group <- gl(2,10,20, labels=c("Ctl","Trt"))
weight <- c(ctl, trt)
lm.D9 <- lm(weight ~ group)
lm.D90 <- lm(weight ~ group - 1) # omitting intercept
anova(lm.D9)
summary(lm.D90)
opar <- par(mfrow = c(2,2), oma = c(0, 0, 1.1, 0))
plot(lm.D9, las = 1) # Residuals, Fitted, ...
par(opar)
# Linear mixed model
dataset <- data.frame(y=rnorm(8), x=factor(c(rep(1,4),rep(0,4))), z=factor(rep(c(1,0),4)))
mixlm <- lm(y~x*r(z), data = dataset)
Anova(mixlm,type="III")
### less simple examples in "See Also" above
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