Lecture 9: Confidence Intervals, Residuals
Formal Analysis
Formal Analysis
The Analysis of Variance –
ANOVA
The F Ratio and P-value
only indicate if statistically
significant differences exist.
The F Ratio and P-value do
not tell you which treatment
means are different.
1
Confidence Interval
2
Confidence Interval
Gives a range of values for
the likely size of the
difference between two
population means.
∗
1
1
95% confidence with df = df for
Error
3
Confidence Interval
4
Confidence Interval
df = 18, 95% Confidence
143.1
163.0
2.101 120.05
t*=2.101
1
10
1
10
19.9 2.101 4.9
19.9 10.3
30.2to 9.6
MSError = 120.05
5
6
1
Lecture 9: Confidence Intervals, Residuals
Interpretation
Generalization
We are 95% confident that
the difference in mean
blood pressures for the two
diets is between –30.2
mmHg and –9.6 mmHg.
Men with Stage 1 hypertension
on the 50 mmol Na/day diet will
have an average blood pressure
from 9.6 to 30.2 mmHg lower
then men with Stage 1
hypertension on the 200 mmol
Na/day diet.
7
JMP
8
Oneway Analysis of Systolic BP (mmHg) By Diet
Oneway Anova
t Test
The t Test part of the output
has for the
Dif = 200 mmol – 50 mmol.
200 mmol/day-050 mmol/day
Assuming equal variances
Difference
19.9000 t Ratio
4.061224
Std Err Dif
4.9000 DF
18
Upper CL Dif 30.1945 Prob > |t|
0.0007*
Lower CL Dif
9.6055 Prob > t
0.0004*
Confidence
0.95 Prob < t
0.9996
-20
-10
0
10
20
Upper CL Dif: 30.1945
Lower CL Dif: 9.6055
9
Residuals
10
JMP
We should check the
residuals to see if they
satisfy the Fisher
Conditions.
From the menu on the
output go to Save –
Residuals.
11
12
2
Lecture 9: Confidence Intervals, Residuals
Equal Standard Deviation
Oneway Analysis of Systolic BP
(mmHg) centered by Diet By Diet
25
20
15
10
5
0
Plot the residuals versus
diet.
Compute means and
standard deviations.
-5
-10
-15
-20
-25
050 mmol/day
200 mmol/day
Diet
Means and Std Deviations
Level
050 mmol/day
200 mmol/day
Number
Mean
Std Dev
10
10
5.684e-15
0
11.7516
10.0995
13
Fisher Conditions
14
Normal Distribution
Residuals add to zero.
Residuals have the same standard
deviations.
Analyze the distribution of
residuals to see if they
could have come from a
normal distribution.
 Although not exactly the same, the
two standard deviations are very close
to being equal.
 Only worry if one standard deviation
is 2, 3 or more times as big as another.
15
16
Distributions
Systolic BP (mmHg) centered by Diet
Fisher Conditions
1.64
1.28
0.67
0.0
-0.67
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Histogram is slightly skewed
right.
Box plot is slightly skewed right.
Most points come close to the
diagonal (normal model) line.
0.2
-1.28
0.1
-1.64
10
8
6
4
2
-30
-20
-10
0
10
20
30
17
18
3
Lecture 9: Confidence Intervals, Residuals
Normal Distribution?
Consequence
The reported P-value may not be
the true P-value.
However, the P-value (0.0007) is
so small even if it is not exactly
correct, we would still conclude
that there is a difference in diets.
There is some doubt as to
whether the residuals come
from a normal distribution.
19
20
Consequence
Comment
The stated confidence level for
the confidence interval (95%)
may not be the true
confidence.
The true confidence may be
lower.
The analysis of variance is still
valid and the conclusion that
there is a statistically
significant difference in mean
blood pressure for the two
diets is not in doubt.
21
22
Comment
Comment
The true P-value and true
confidence level may be
slightly different from what
is reported.
Remember the analysis of
residuals is a secondary analysis
and usually doesn’t make a
difference unless the conditions
are severely violated and the
analysis is marginal (a P-value
close to 0.05).
23
24
4