The research question specifies the average as the statistical
parameter of interest. The sample mean for creatinine level is 2.04
and is quite close to the hypothesized value of two. A one-sample
t test is an appropriate method to determine if the observed sample
mean is significantly different from hypothesized value or if the
difference can be attributed to sampling error. In this research question,
the null hypothesis is that the true mean creatinine level equals
two. A one-sided alternative (greater than) corresponds to the research
question. Alternative hypotheses are in two forms, one-sided and
two-sided; the choice depends on how the research question is posed.
Two-sided alternatives are appropriate when testing for equality
with the hypothesized value. One-sided alternatives apply when discovering
a difference from the hypothesized value in only one direction (greater
than or less than) is of interest.
The p-value is the key result from a hypothesis test.
Since we are interested in determining if the mean creatinine level
exceeds 2, “Prob > t” is the correct p-value. A
p-value expresses the likelihood of obtaining the sample mean or something
more extreme assuming the null hypothesis is true. Small p-values
cause a rejection of the null hypothesis. The significance level
is the risk of rejecting the null hypothesis when it is in fact true.
A p-value less than the chosen significance level means that the
null hypothesis is rejected. For this test we’ll choose a
significance level of 0.05. The p-value of 0.0299 is less than 0.05,
hence the null hypothesis is rejected in favor of the alternative.
The test tells us that the mean creatinine level is significantly
larger than 2.
In
the Normal quantile plot, the line corresponds to the Normal distribution
that best fits the data. The normality assumption is satisfied to
the extent that the observations lie close to this line. The observed
Creatinine values show considerable departure from the line. The JMP
Distribution platform also provides the Shapiro-Wilk normality test
as an option. From the Creatinine drop-down select Continuous Fit
> Normal and the Fitted Normal output will appear. From the Fitted
Normal drop-down select Goodness of Fit.
Figure 6.8 Normal Quantile Plot and Shapiro-Wilks Normality Test Output for Creatinine shows the
corresponding output. The p-value of <0.0001 causes a reject of
the null hypothesis that the data is normally distributed. Hence,
the normality assumption is not satisfied for the one-sample t-test
and an alternative should be sought.
Often in colloquial use, the word “average”
implies a measure of centrality for a distribution rather than the
sample mean. In a skewed distribution, the median is often a better
measure of centrality. The JMP Distribution > Test Mean feature
offers a non-parametric alternative to the one-sample t-test, the
Wilcoxon Signed Rank, which tests the median against a hypothesized
value. The Wilcoxon Signed Rank test requires the data to be symmetrically
distributed, which is not the case with Creatinine, so this test is
not appropriate.
The column “Creatinine > 2” allows us
to estimate the proportion of patients that have creatinine levels
greater than 2. From
Figure 6.5 Histograms and Summary Statistics from Distribution Platform,
we find this proportion to be 0.637. In a sample from a distribution
with a median creatinine level of 2, we would expect the proportion
having a creatinine greater than two to be 0.50. A hypothesis test
for a proportion can be used to determine if our proportion of 0.637
is significantly greater than 0.50. If this is the case, then we
can conclude that the median creatinine is significantly larger than
two.
The p-value of <0.0001
is well below the 5% significance level. Hence the null hypothesis
that the proportion of patients with creatinine level greater than
two is 0.50 is rejected in favor of the alternative that the proportion
is greater than 0.50. We conclude that the median creatinine level
is significantly greater than two.