Often, the impact of an assumption violation on the multiple linear
regression result depends on the extent of the violation (such as the how
inconstant the variance of Y is, or how skewed
the Y population distribution
is). Some small violations may have little practical effect on the analysis,
while other violations may render the multiple linear regression result
uselessly incorrect or uninterpretable.
Apparent lack of independence
in the fitted Y values may be caused by the existence of an implicit X
variable in the data, an X variable that was not explicitly used in the linear
model. In this case, the best model may still be linear, but may not include
all the original X variables. If there is a linear trend in the plot of the
regression residuals against the fitted values, then an implicit X variable
may be the cause. A plot of the residuals against the prospective new X
variable should reveal whether there is a systematic variation; if there is,
you may consider adding the new X variable to the linear model.
A "new" X variable might be derived from one or more X variables already in
the equation, such as using the square of X1 along with X1 to handle curvature
in X1, or adding X1*X2 as a new variable to handle interaction
between X1 and X2.
If an implicit X variable is not included in the fitted model, the fitted
estimates for the coefficients may be biased,
and not very meaningful, and the fitted Y values may not be accurate.
Another possible cause of apparent dependence between the Y observations is
the presence of an implicit block
effect. (The block effect can be considered another type of implicit X
variable, albeit a discrete one.) If a blocking variable is suspected, an analysis
of covariance can be performed, essentially dividing the data into
different regression equations based on the value of the blocking variable.
If multiple values of Y are collected at the same values of X, this can act
as another type of blocking, with the unique combinations of values of the Xs
acting as blocks. These multiple Y measurements may be less variable than the
overall variation in Y, and, given their common values of the Xs, they are not
truly independent
of each other. If there are many replicated X values, and if the variation
between Y at replicated values is much smaller than the overall residual
variance, then the variance of the estimate of the coefficients may be too
small, making the test of whether they are 0 (and, the test of the goodness of
the overall fit) anticonservative (more likely than the stated significance
level to reject the null
hypothesis, even when it is true). In this case, an alternative method is
to replace each replicated unique combination of X values by a single data
point with the average Y value, and then perform the regression analysis with
the new data set. A possible drawback to this method is that by reducing the
number of data points, the degrees of freedom associated with the residual
error is reduced, thus potentially reducing the power
of the test.
Whether the Y values are independent
of each other is generally determined by the structure of the experiment from
which they arise. Y values collected over time may be serially correlated
(here time is the implicit factor). If the data are in a particular order,
consider the possibility of dependence. (If the row order of the data reflect
the order in which the data were collected, an index
plot of the data [data value plotted against row number] can reveal
patterns in the plot that could suggest possible time effects.)
For serially correlated error terms, the estimates of the coefficients will
be unbiased,
but the estimates of their variances will not be reliable. If they are
positively serially correlated, the estimate of residual variance and the
estimates of the variances of the coefficients may all be too small, making
the tests and confidence intervals that involve them unreliable. This kind of
serial correlation may appear when there are one or more implicit
X variables.
If you are unsure whether your Y values are independent, you may wish to
consult a statistician or someone who is knowledgeable about the data
collection scheme you are using.
If two or more of the X variables are nearly linear combinations of each
other, the X variables are multicollinear. In this situation, you may be able
to find a good multiple linear fit for Y, but the values of the individual
coefficients may be highly variable. Thus, you might be able to predict Y with
reasonable accuracy, but you would not be able to draw any reliable
conclusions about the coefficients. And the fitted coefficients can vary
widely from sample to sample of data, or if a single X variable is added or
deleted from the equation. Various formal and informal diagnostics
may help detect multicollinearity. There are also some methods
designed to deal with multicollinearity.
In cases of severe multicollinearity, it may
not be possible to calculate some of the diagnostic measures of influence
or leverage,
or even to perform the fit itself. In such cases, the data are said to be
ill-conditioned.
Values may not be identically distributed because of the presence of outliers.
Outliers are anomalous values in the data. Outliers may have a strong
influence over the fitted coefficients, giving a poor fit to the bulk of the
data observations. Outliers tend to increase the estimate of residual
variance, lowering the chance of rejecting the null
hypothesis. They may be due to recording errors, which may be correctable,
or they may be due to the Y values not all being sampled from the same
population. Apparent outliers may also be due to the Y values being from the
same, but nonnormal,
population. Outliers may show up clearly in a scatterplot of Y and one of the
X variables, as points that do not lie near the general trend of the data.
However, a point may be an unusual value in either X or Y without necessarily
being an outlier in the scatterplot.
Once the regression line has been fitted, the boxplot
and normal
probability plot (normal Q-Q plot) for residuals may suggest the presence
of outliers in the data. After the fit, outliers are usually detected by
examining the residuals
or the high-leverage
points.
The method of least squares involves minimizing the sum of the squared
vertical distances between each data point and the fitted line. Because of
this, the fitted line can be highly sensitive to outliers.
(In other words, least squares regression is not resistant
to outliers, and thus, neither are the fitted coefficient estimates.) An
outlier may act as a high-leverage
point, distorting the fitted equation and perhaps fitting the main body of
the data poorly.
If you find outliers in your data that are not due to correctable errors,
you may wish to consult a statistician as to how to proceed.
The values in a sample may indeed be from the same population, but not
from a normal one. Signs of nonnormality
are skewness
(lack of symmetry) or light-tailedness
or heavy-tailedness.
The boxplot,
histogram,
and normal
probability plot (normal Q-Q plot), along with the normality test, can
provide information on the normality of the population distribution. However,
if there are only a small number of data points, nonnormality can be hard to
detect. If there are a great many data points, the normality test may detect
statistically significant but trivial departures from normality that will have
no real effect on the multiple linear regression's tests (since, for example,
the t statistic for the test of a coefficient will converge in probability to
the standard normal distribution by the law of large numbers).
For data from a normal distribution, normal probability plots should
approximate straight lines, and boxplots should be symmetric (median and mean
together, in the middle of the box) with no outliers.
Except for substantial nonnormality that leads to outliers
in the X-Y data, if the number of data points is not too small, then the
multiple linear regression statistic will not be much affected even if the
population distributions are skewed.
Robust
statistical tests operate well across a wide variety of distributions. A test
can be robust for validity, meaning that it provides P values close to the
true ones in the presence of (slight) departures from its assumptions. It may
also be robust for efficiency, meaning that it maintains its statistical power
(the probability that a true violation of the null
hypothesis will be detected by the test) in the presence of those
departures. Linear regression is fairly robust for validity against
nonnormality, but it may not be the most powerful test available for a given
nonnormal
distribution, although it is the most powerful
test available when its test assumptions are met. In the case of nonnormality,
a non-least-squares
regression method, or employing a transformation
of one or more X variables may result in a more powerful test.
If the variance of the Y is not constant, then the the error variance will
not be constant. The most common form of such heteroscedasticity
in Y is that the variance of Y may increase as the mean of Y increases, for
data with positive X and Y.
Unless the heteroscedasticity of the Y is pronounced, its effect will not
be severe: the least squares estimates will still be unbiased,
and the estimates of the coefficients will either be normally distributed if
the errors are normally
distributed, or at least normally distributed asymptotically (as the
number of data points becomes large) if the errors are not normally
distributed. The estimate for the variance of the coefficients will be
inaccurate, but the inaccuracy is not likely to be substantial if the X values
are symmetric about their means.
Heteroscedasticity of Y is usually detected informally by examining
the X-Y scatterplots of the data before performing the regression. If both
nonlinearity and unequal variances are present, employing a transformation
of Y may have the effect of simultaneously improving the linearity and
promoting equality of the variances. Otherwise, a weighted
least squares multiple linear regression may be the preferred method of
dealing with nonconstant variance of Y.
If the linear model is not the correct one for the data, then the
coefficient estimates and the fitted values from the multiple linear
regression will be biased,
and the fitted coefficient estimates will not be meaningful. Over a restricted
range of X or Y, nonlinear models may be well approximated by linear models
(this is in fact the basis of linear interpolation), but for accurate
prediction a model appropriate to the data should be selected. An examination
of the X-Y scatterplots may reveal whether the linear model is
appropriate. If there is a great deal of variation in Y, it may be difficult
to decide what the appropriate model is; in this case, the linear model may do
as well as any other, and has the virtue of simplicity.
The usual multiple linear regression model assumes that the observed X
variables are fixed, not random. If the X values are are not under the control
of the experimenter (i.e., are observed but not set), and if there is in fact
underlying variance in the X variables, but they have the same variance, the
linear model is called the errors-in-variables model or the
structural model. The least squares fit will still give the best linear
predictor of Y, but the estimates of the coefficients will be biased.
If the assumption of the linear model is correct, the plot of the observed
Y values against X should suggest a linear band across the graph. Outliers
may appear as anomalous points in the graph, often in the upper righthand or
lower lefthand corner of the graph. (A point may be an outlier in either X or
Y without necessarily being far from the general trend of the data.)
If the linear model is not correct, the shape of the general trend of the
X-Y plot may suggest the appropriate function to fit (e.g., a polynomial,
exponential, or logistic function). Alternatively, the plot may suggest a
reasonable transformation
to apply. For example, if the X-Y plot arcs from lower left to upper right so
that data points either very low or very high in X lie below the equation
suggested by the data, while the data points with middling X values lie on or
above that straight line, taking square roots or logarithms of the X values
may promote linearity.
If the assumption of equal variances for the Y is correct, the plot of the
observed Y values against X should suggest a band across the graph with
roughly equal vertical width for all values of X. (That is, the shape of the
graph should suggest a tilted cigar and not a wedge or a megaphone.)
A fan pattern like the profile of a megaphone, with a noticeable flare
either to the right or to the left as shown in the picture suggests that the
variance in the values increases in the direction the fan pattern widens
(usually as the sample mean increases), and this in turn suggests that a transformation
of the Y values may be needed.
Unfortunately, simple X-Y plots may not be as useful in multiple regression
as they are for simple
linear regression. If there is multicollinearity,
then that can cause the plots of Y against individual X values to be
misleading. For example, the apparent increase in variance for Y as X1
increases might be due to the effect of other X variables on Y.
If the number of data points is small, it may be difficult to detect
assumption violations. With small samples, violation assumptions such as nonnormality
or heteroscedasticity
of variances are difficult to detect even when they are present. With a
small number of data points multiple linear regression offers less protection
against violation of assumptions. With few data points, it may be hard to
determine how well the fitted equation matches the data, or whether a
nonlinear function would be more appropriate.
If the ratio of the total number of coefficients (including the intercept)
to the total number of data points is greater than 0.4, it will often be
difficult to fit a reliable model. Many of the individual data points may
become influential
points, because there is so little information (data) available for each
coefficient to be fitted.
A rule of thumb is to aim to have the number of data points be at least 6
times, and ideally at least 10 times, the number of X variables.
Even if none of the test assumptions are violated, a linear regression on a
small number of data points may not have sufficient power
to detect a significant difference between a coefficient and 0, even if the
coefficient is non-zero. The power depends on the residual error, the observed
variation in X, the selected significance (alpha-) level of the test, and the
number of data points. Power decreases as the residual variance increases,
decreases as the significance level is decreased (i.e., as the test is made
more stringent), increases as the variation in observed X increases, and
increases as the number of data points increases. If a statistical
significance test with a small number of data values produces a surprisingly
non-significant P
value, then lack of power may be the reason. The best time to avoid such
problems is in the design stage of an experiment, when appropriate minimum
sample sizes can be determined, perhaps in consultation with a statistician,
before data collection begins.
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