Download Assumptions underlying regression analysis

Assumptions underlying
regression analysis
(Session 05)
SADC Course in Statistics
Learning Objectives
At the end of this session, you will be able to
• describe assumptions underlying a
regression analysis
• conduct analyses that will allow a check on
model assumptions
• discuss the consequences of failure of
assumptions
• consider remedial action when assumption
fail
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Checking assumptions
Describing the relationship carries no
assumptions.
However, inferences concerning the slope of
the line, e.g. by use of t-tests or F-tests,
are subject to certain assumptions.
Checking assumptions is important to avoid
pitfalls associated with making invalid
conclusions.
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Assumptions
The simple linear regression model is:
yi
= 0
+ 1 xi
+ i
In addition to assuming a linear form for the
model, the i are assumed to be
• independent, with
• zero mean and constant variance 2,
• and be normally distributed.
Note: Model predictions, often called fitted
values, are
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How to check assumptions?
The usual approach is to conduct a residual
analysis. Residuals are deviations of
observed values from model fitted values.
Paddy
data
relating
yield to
fertiliser
Residual
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Residual Plots
Plotting residuals in various ways allows
failure of assumptions to be detected.
e.g. To check the normality assumption,
• plot a histogram of the residuals (provided
there are enough observations);
• or do a normal probability plot of residuals.
A straight line plot indicates that the
normality assumption is reasonable.
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Residual Plots - continued
Most useful is a plot of residuals against fitted
values( yˆ i ) .
It helps to detect failure of the variance
homogeneity assumption. Also helps to
identify potential outliers.
e.g. If standardised residuals are used,
i.e. residuals/standard error,
then 95% of observations would be expected
to lie between –2 and +2.
A random scatter with no obvious pattern is
good! Some examples follow….
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Some Residual Plots
x
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A random scatter as above is good.
It shows no obvious departures of the
variance homogeneity assumption.
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Some Residual Plots
x
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Variance increases with increasing x.
Could try a loge(y) transformation.
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Some Residual Plots
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Indication that the response is a binomial
proportion. Use a logistic regression model.
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Some Residual Plots
x
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x x
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x x
x x
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Lack of linearity. Pattern indicates an
incorrect model - probably due to a missing
squared term.
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Some Residual Plots
x
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Presence of an outlier. Investigate if
there is a reason for this odd-point.
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Consequences of assumption failure
Studies on consequences of assumption
failure have demonstrated that:
• tests and confidence intervals for means
are relatively robust to small departures
from non-normality;
• the effects of non-homogeneous variance
can be large, but not so serious if sample
sizes in different sub-groupings are equal;
• dependence of observations can badly
affect F-tests.
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Dealing with assumption failure
One approach is to find a transformation
that will stabilize the variance.
Some typical transformations are:
• taking logs (useful when there is skewness);
• square root transformation;
• reciprocal transformation.
Sometimes theoretical grounds will determine
the transformation to use, e.g. when data are
Poisson or Binomial. However, in such cases,
exact methods of analysis will be preferable.
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Dealing with non-independence
The assumption of independence is quite
critical. Some attention to this is needed at
the data collection stage.
If observations are collected in time or space,
plotting residuals in time (or space) order
may reveal that subsequent observations are
correlated.
Techniques similar to those used in time
series analysis or analysis of repeated
measurements data may be more
appropriate.
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An illustration – using paddy data
Histogram of standardised residuals after
fitting a linear regression of yield on fertiliser.
This is a check on the normality assumption.
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A normal probability plot…
Another check on the normality assumption
Do you
think the
points
follow a
straight
line?
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Std. residuals versus fitted values
Checking assumption of variance
homogeneity, and identification of outliers:
Do you
judge this
to be a
random
scatter?
Are there
any
outliers?
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Conclusion:
The residual plots showed no evidence of
departures from the model assumptions.
We may conclude that fertiliser does
contribute significantly to explaining the
variability in paddy yields.
Note: Always conduct a residual analysis
after fitting a regression model. The same
concepts carry over to more complex models
that may be fitted.
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Practical work follows to ensure
learning objectives are
achieved…
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