A variable does not have a t or F statistic in isolation. If you change
what other variables are in the model, you will change its significance. If
things two X-variables are sufficiently related to each other, either one
may explain a Y-variables, but once either one is in the model you won't
need the other. You can also find situations where neither is significant
alone, but both a significant together.
When you look at an ANOVA table, it is also important to know whether its
author is using sequential or partial sums of squares. For example, say you
have something like
Term sum of squares
x1 some number
x2 some number
If the author is tabulating sequential sums of squares, the SS for x1 will
be with x2 NOT in the model and the SS of x2 will be with x1 IN the model
(because x1 precedes x2 in the table). If he tabulates sequential sums of
squares this way:
Term sum of squares
x2 some number
x1 some number
the numbers will be different, but their sums will be the same in both
cases.
If the author is tabulating partial sums of squares, the SS for x1 will be
with x2 IN the model, and the SS for x2 will be with x1 in the model. In
that case the order won't matter.
I can't explain the whole thing in a short E-mail. A good textbook chapter
on "model selection" would help. If you are using JMP software, the chapter
on "Multiple Regression" in "JMP Software: ANOVA and Regression" (SAS
Institute, 2003) is good. (I admit to bias on that last point. I'm one of
their contract instructors.)
-------------------------------------------
Emil M Friedman, PhD
2304 Richmond Road
Beachwood, OH 44122
[log in to unmask]
216-591-1750 (voice)
775-249-6744 (fax)
----- Original Message -----
From: "Kim Pearce" <[log in to unmask]>
To: <[log in to unmask]>
Sent: Thursday, July 31, 2003 7:44 AM
Subject: Multiple Regression: coefficient significance
> Dear all,
>
> I would like to ask a query about the testing of model coefficients in a
> multiple regression. According to Montgomery and Peck, 1992 (p138) there
> are two methods of assessing significance. I would like to know the
> difference between the two. Can anyone shed some light? I have provided
an
> example below.
>
> 1) M & P firstly use the p values associated with the T statistics. Here
we
> are testing the significance of any individual regression coefficient,j,
> i.e. H_O: B_j =0 vs H_1:B_j ne 0 .
> M &P say that "this is a partial or marginal test because the regression
> coefficient B_j depends on all the other regressor variables in the model.
> Thus this is a test of the contribution of x_j given the other regressors
in
> the model." So in my example below we could say that the coefft of my
> variable 'radio' is non significant (as is magazine) and we could delete
> radio from the model as it has the largest non sig p value...we could then
> regenerate the model using 'TV' and 'magazine' and assess their
significance
> again...this procedure is like 'backward elimination'.
>
> 2) M & P also go on to talk about the 'extra sums of squares'
method...which
> is "determining the contribution to the regression sums of squares of x_j
> given the other regressors are included in the model" (this seems similar
to
> what they have said in 1. above??). They test the contribution of an
> additional variable using:
>
> SeqSS for that variable / MSE of the full regression model
>
> In the example below, for the radio variable, we would thus have
28.92/19.52
> = 1.48 tested against the F distribution with 1 & 6 degress of freedom.
> This is actually equivalent to the T statistic corresponding to the
variable
> 'radio' (as T^2 = F) and it is assessing the significance of 'radio' given
> that 'TV' and 'magazine' are in the model.
>
> My questions are:
>
> A) For method 2, could we also use the SeqSS from the *same output*
> corresponding to 'TV' and the sequential SS from the *same output*
> corresponding to 'magazine' to assess the significance of these two
> variables...so we would have:
>
> H_0: B_TV=0 vs B_TV ne 0 (given magazine and radio already in the model)
> 991.57/19.52 = 50.8 tested against F_1,6
>
> and
>
> H_O: H_0: B_magazine=0 vs B_magazine ne 0 (given TV and radio already in
the
> model)
> 174.04/19.52 = 8.92 tested against F_1,6
>
> If we can do the above then 'TV' is significant and 'magazine' is
> significant (the latter is non significant using the T statistics - why is
> this?). Also if we can do this - which is the better to use -> T
statistics
> or the seqSS procedure??
>
> If we construct a model so that the 3 variables enter the model in a
> different order, the T statistics (and associated p values) do not change
> but the seq SS will, of course, change thus if we can use this 'Seq SS'
> method to evaluate the significance of all 3 coeffts then, on using the
Seq
> SS method, couldn't we see the significance of the 3 coefficients changing
> depending on which order they entered the model?
>
> or
> B) Do we only use method 2 to assess the significance of the *final*
> variable in the model (in my case 'radio')? In which case the SeqSS/MSE
> gives a result which is equivalent to the T statistic?
>
> (M&P only provide an example which deals with the case B scenario above).
>
> I have always only used T-statistics as a quick way of evaluating
> significance of coefficients; the SeqSS technique is puzzling me!
>
> Many thanks for your help,
> Kim.
>
> The regression equation is
> y = 266 + 6.73 TV + 3.26 magazines + 4.51 radio
>
> Predictor Coef StDev T P
> Constant 266.23 16.34 16.29 0.000
> TV 6.727 1.344 5.01 0.002
> magazine 3.257 1.642 1.98 0.095
> radio 4.507 3.703 1.22 0.269
>
> S = 4.418 R-Sq = 91.1% R-Sq(adj) = 86.6%
>
> Analysis of Variance
>
> Source DF SS MS F P
> Regression 3 1194.53 398.18 20.40 0.002
> Residual Error 6 117.11 19.52
> Total 9 1311.64
>
> Source DF Seq SS
> TV 1 991.57
> magazine 1 174.04
> radio 1 28.92
>
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