1 General Purpose
Trend tests are wonderful, because they
provide markedly better sensitivity for demonstrating incremental
effects from incremental treatment dosages, than traditional
statistical tests. In the Chap. 15 trend tests for continuous outcome
data are reviewed. In the current chapter trend tests for binary
outcome data are assessed.
2 Schematic Overview of Type of Data File
3 Primary Scientific Question
Do incremental dosages of a medicine
cause incremental numbers of patients to become responders.
4 Data Example
In a 106 patient study the primary
scientific question was: do incremental dosages of an
antihypertensive drug cause incremental numbers of patients to
become normotensive. The entire data file is in
extras.springer.com, and is entitled “chapter40trendbinary”.
|
Responder
|
Treatment
|
|
1,00
|
1,00
|
|
1,00
|
1,00
|
|
1,00
|
1,00
|
|
1,00
|
1,00
|
|
1,00
|
1,00
|
|
1,00
|
1,00
|
|
1,00
|
1,00
|
|
1,00
|
1,00
|
|
1,00
|
1,00
|
|
1,00
|
1,00
|
|
1,00
|
2,00
|
5 A Contingency Table of the Data
The underneath contingency table shows
that with incremental dosages the odds of responding rises from
0.67 to 1.80.
|
Dosage 1
|
Dosage 2
|
Dosage 3
|
|
|
Numbers responders
|
10
|
20
|
27
|
|
Numbers non-responders
|
15
|
19
|
15
|
|
Odds of responding
|
0.67(10/15)
|
1.11(20/19)
|
1.80(27/15)
|
First, we will try and summarize the
data in a graph. Start by opening the data file in SPSS.
6 3-D Bar Charts
Command:
-
Graphs....Legacy Dialogs....3-D Bar Charts....X-axis represents....mark Groups of cases....Z-axis represents....mark Groups of cases....click Define....X Category Axis: treatment....Z Category Axis: responders....click OK.
The above graph is shown in the output
sheets. The treatment-1-responder-0 bar is invisible.
Command:
-
Double-click the graph in order to activate it....“Chart Editor” comes up....click Rotating 3-D chart....3-D Rotation....Horizontal: enter 125....the underneath graph comes up showing the magnitude of the treatment-1-responder-zero bar.
The above two graphs show, that
incremental treatment dosages of an antihypertensive drug seem to
cause incremental numbers of responders (patients becoming
normotensive). However, the numbers of non-responders are the
controls, and their pattern is, equally, important. We, first, will
perform a multiple groups chi-square test in order to find out,
whether there is any significant difference in the data.
7 Multiple Groups Chi-Square Test
For analysis the statistical model
Crosstabs in the module Descriptive Statistics is required.
Command:
-
Analyze....Descriptive Statistics....Crosstabs....Row(s): responder....Column(s): treatment....Statistics....Chi-Square Tests....click OK.
Chi-square tests
|
Value
|
df
|
Asy mp. Sig. (2-sided)
|
|
|---|---|---|---|
|
Pearson chi-square
|
3,872a
|
2
|
,144
|
|
Likelihood ratio
|
3,905
|
2
|
,142
|
|
Linear-by-linear association
|
3,829
|
1
|
,050
|
|
N of valid cases
|
106
|
The above table shows that, indeed,
the Pearson chi-square value for multiple groups testing is not
significant with a chi-square value of 3,872 and a p-value of
0,144, and we have to conclude that there is, thus, no significant
difference between the odds of responding to the three
dosages.
8 Chi-Square Test for Trends
Subsequently, a chi-square test for
trends can be executed, a test, that, essentially, assesses,
whether the above odds of responding (number of responder/numbers
of non-responders per treatment group) increase significantly. The
“linear-by-linear association” from the same table is appropriate
for the purpose. It has approximately the same chi-square value,
but it has only 1 degree of freedom, and, therefore, it reaches
statistical significance with a p-value of 0,050. There is, thus, a
significant incremental trend of responding with incremental
dosages.
Chi-square tests
|
Value
|
df
|
Asy mp. Sig. (2-sided)
|
|
|---|---|---|---|
|
Pearson chi-square
|
3,872a
|
2
|
,144
|
|
Likelihood ratio
|
3,905
|
2
|
,142
|
|
Linear-by-linear association
|
3,829
|
1
|
,050
|
|
N of valid cases
|
106
|
The trend in this example can also be
tested using logistic regression with responding as outcome
variable and treatment as independent variable (enter the latter as
covariate, not as categorical variable).
9 Conclusion
Trend tests provide markedly better
sensitivity for demonstrating incremental effects from incremental
treatment dosages, than traditional statistical tests. In the Chap.
16 trend tests for continuous outcome
data are reviewed. In the current chapter trend tests for binary
outcome data are assessed.
10 Note
More background, theoretical, and
mathematical information of trend testing is given in Statistics
applied to clinical studies 5th edition, Chap. 27, Springer
Heidelberg Germany, 2012, from the same authors.