1 General Purpose
In clinical studies the outcome is
often assessed with numbers of responders and nonresponders to some
treatment. If the proportion of responders is statistically
significantly larger than zero, then the treatment is
efficaceous.
2 Schematic Overview of Type of Data File
3 Primary Scientific Question
Is the proportion of responders
significantly larger or smaller than zero.
4 Data Example
Underneath are the first 10 patients of
a 55 patient file of patients responding to hypertensive treatment
or not. We wish to test whether the number of patients who respond
is significantly larger than a number of 0.
|
Outcome
|
|
0
|
|
0
|
|
0
|
|
0
|
|
0
|
|
0
|
|
0
|
|
0
|
5 Analysis: One-Sample Z-Test
The 55 patient data file is in
extras.springer.com, and is entitled “chapter 34onesamplebinary”.
Open it in SPSS.
Command:
-
Analyze....Descriptive statistics....Descriptives....Variable(s): responder....Options: mark: mean, sum, SE, mean....click Continue....click OK.
Descriptive statistic
|
N
|
Sum
|
Mean
|
||
|---|---|---|---|---|
|
Statistic
|
Statistic
|
Statistic
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Std. error
|
|
|
afdeling
|
55
|
20,00
|
,3636
|
,06546
|
|
Valid N (listwise)
|
55
|
|||
The z-value as obtained equals
0,3636/0,06546 = 5,5545. This value is much larger than 1,96, and,
therefore, the nullhypothesis of no difference from 0 can be
rejected at p < 0,001. A proportion of 20/55 is significantly
larger than 0.
6 Alternative Analysis: Binomial Test
If the data do not follow a Gaussian
distribution, this method will be required, but, with Gaussian
distributions, it may be applied even so. For analysis the
statistical model One Sample in the module Nonparametric Tests is
required.
Command:
-
Analyze....Nonparametric Tests....click One Sample....click Fields....Test Fields: enter “responder”....click Settings....click Choose Tests....mark Customize tests ....mark Compare observed binary probability…(Binomial test)....click Options ....Hypothesized proportion: enter 0,00....click OK.....a warning comes up: SPSS does not accept 0,00....click Fix....replace 0,00 with 0,00001....click OK....click Run.
The underneath table is in the output.
The proportion observed is significantly different from 0,00 at
p < 0,0001. This result is similar to that of the above z-test.
Hypothesis test summary
7 Conclusion
The significant results indicate that
the nullhypothesis of no effect can be rejected. The proportion of
responders is significantly larger 0,00. It may be prudent to use
nonparametric tests, if normality is doubtful, like in the small
data example given.
8 Note
The theories of nullhypotheses and
frequency distributions for binary outcome data are reviewed in
Statistics applied to clinical studies 5th edition, Chap. 3,
Springer Heidelberg Germany, 2012, from the same authors.