Transcript Inferences about the regression line

Inferences about
the regression line
(Session 03)
SADC Course in Statistics
Learning Objectives
At the end of this session, you will be able to
• make inferences concerning the slope of
the regression line
– through the use of a t-test
– using an analysis of variance F-test
• describe and interpret the components of
an anova table
• explain the meaning of s2 in the analysis of
variance and the importance of attention to
the corresponding degrees of freedom
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Smoking and death rates again!
We consider again the example used in the
previous session concerning the average
number of cigarettes smoked per adult in
1930 and the death rate per million in 1952
for sixteen countries.
Previously we described this relationship.
We now ask whether this relationship is a
real one, or whether it could be just a chance
occurrence.
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Recall model estimates
-----------------------------------------------------deathrate|Coef. Std.Err.
t
P>|t| [95% Conf.Int.]
---------+-------------------------------------------cigars
| .2410
.0544
4.43 0.001
.1245
.3577
const.
| 28.31
46.92
0.60 0.556 -72.34 128.95
------------------------------------------------------
Estimates ̂ and ̂ of unknown parameters 
and  of the model y =  +  x + 
Estimated equation is: ŷ = 28.31 + 0.241 * x
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Assessing the regression line
Is there a real relationship between y and x?
In the model y = +x, need to test the
hypothesis:
H0: no linear relationship, i.e. slope  = 0
H1: y is linearly related to x, i.e. slope   0
One approach is to use a t-test, i.e. first
calculate t below.
slope - 0
0.241
t

 4.43
s.e.(slope) 0.0544
(Same as t-value for “cigars” in slide 4)
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Interpreting results about the slope
Compare calculated t of 4.43 with tabulated
t-value with 14 d.f.
The 2-sided tabulated value is 2.98 at a 1%
significance level, and 4.14 at a 0.1% sig. level.
It may be concluded that there is strong evidence
to reject the null hypothesis H0.
i.e. there is strong evidence of a linear relationship
between smoking and death rates.
Note: In practice, just the computer output P>|t| ,
will be interpreted. This is the p-value for the test.
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Another approach…
The same hypothesis as above can also be
tested using an analysis of variance (ANOVA)
This involves splitting the overall variation in
y into two components:
• Variation due to the regression, i.e. due to the
presence of the explanatory variable x
• Balance (or residual) variation, i.e. variation that
is not explained by the explanatory variable
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400
500
Deviations from overall mean
300
Deviation from mean
0
100
200
Mean
=215
0
500
1000
Cigarettes smoked (x)
1500
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2000
8
500
Deviations from regression and
residual deviation
400
Residual deviation
0
100
200
300
Deviation from
regression
0
500
1000
Cigarettes smoked (x)
Death rate (y)
1500
2000
Fitted values
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Analysis of Variance (ANOVA)
Source
Regression
d.f.
1
S.S.
M.S.
F
Prob.
132934.7 132934.7
19.7
0.0006
Residual
14
94637.0
6759.8
Total
15
227571.8
15171.5
ANOVA shows breakdown of total variation into
•
Variation due to regression, and
•
Residual variation
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Analysis of Variance (ANOVA) ctd…
Source
Regression
d.f.
1
S.S.
M.S.
F
Prob.
132934.7 132934.7
19.7
0.0006
Residual
14
94637.0
6759.8
Total
15
227571.8
15171.5
• Mean square (M.S.)=Sum of squares (S.S.)
degrees of freedom(d.f.)
• Need sufficient d.f. for residual M.S. for
reliable significance testing
• Regression has 1 d.f. because 1 slope is being
estimated
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Interpretation Residual Mean Square
• Residual Mean Square (s2) estimates the
underlying variation (2) in y that is not
explained by the x variable
• It is used in the calculation of standard
errors of model estimates (& other estimates
derived from the model)
• Hence it plays a role in determining the
precision of such estimates
• For a simple linear regression model, the
residual degrees of freedom = n – 2.
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Interpretation of Anova table
Significance test:
H0: no linear relationship between death rate
and number of cigarettes smoked (=0)
H1: there is a linear relationship (0)
• F-value of 19.7
• Compare with F-distribution with (1,14) df
• Highly significant: p-value=0.0006
Conclusion: there is a strong evidence of a
linear relationship between death rates and
number of cigarettes smoked.
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ANOVA versus t-test
In our example, anova and t-test were testing
the same hypothesis, so conclusions identical!
However, note that
• the anova can be extended to include more
than one regressor variable
• The t-test can be used to test general
hypotheses concerning the slope,
e.g. H0: slope=1 for testing if a new, simpler
poverty index behaves similarly to a
standard measure previously used.
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Practical work follows to ensure
learning objectives are
achieved…
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