To become familiar with the medical status and demographic characteristics of the hospitalized patients represented in our data set, we begin by creating histograms and descriptive statistics for all variables (except for the unique patient identification numbers) using the Distribution platform. Figure 6.5 Histograms and Summary Statistics from Distribution Platform shows the resulting JMP output.

The creatinine distribution is left-skewed; 63.7% of the patients have creatinine levels greater than 2 mg/dL. The mean creatinine level is 2.04 while the median is 2.15 with values ranging from 0.62 to 2.50 mg/dL. The mean creatinine level is less than the median due to the left skew. Age and length of stay show fairly uniform distributions. The race distribution for this data set is 73% African-American, 20% Asian/Other, and 7% Caucasian. For these patients, 7% have diabetes, 15% have coronary artery disease, and 25% have acute kidney injury.

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 one-sample t-test can be conducted from the JMP Distribution platform. From the drop-down menu select Test Mean and complete the dialog as shown in Figure 6.6 Test Mean Dialog for Creatinine Level.

The Test Mean output is added to the default Distribution output as shown in Figure 6.7 Test Mean Output for Creatinine Level.

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.

The assumptions of a one-sample t-test are random sampling and that the population is normally distributed. Creatinine does not appear normally distributed based on visual inspection of the histogram in Figure 6.7 Test Mean Output for Creatinine Level. The normality assumption for Creatinine can be formally assessed from the JMP Distribution platform in two different ways. Figure 6.8 Normal Quantile Plot and Shapiro-Wilks Normality Test Output for Creatinine shows a normal quantile plot for Creatinine obtained from the Creatinine drop-down menu.

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.

To conduct this test of proportion, select Analyze > Distribution and enter the column “Creatinine > 2” in the Y, Columns field and click Ok. In the resulting dialog, select “Test Probabilities” from the “Creatinine > 2” drop-down window as shown in Figure 6.9 Completed Dialog for Creatinine > 2 Test of Proportion.

Click Done to obtain the results shown in Figure 6.10 JMP Output for Test of Proportion.

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.

The point estimate for the proportion of patients that have a creatinine level greater than 2 (Stage 1 kidney insufficiency) is found in Figure 6.5 Histograms and Summary Statistics from Distribution Platform to be 0.637. Calculating a confidence interval for the proportion will provide an interval estimate which takes into account the precision with which the proportion was estimated. Figure 6.11 Calculating a 95% Confidence Interval for the Proportion of Patients with Creatinine Level > 2 shows how to request a 95% confidence interval from the Distribution platform.

Figure 6.12 Confidence Interval Output from JMP Distribution Platform shows the confidence interval output. Confidence intervals are given for both the proportion of patients that exhibit creatinine levels consistent with Stage 1 kidney insufficiency and those whose creatinine levels do not indicate kidney insufficiency.

The 95% confidence interval for the proportion of patients with Creatinine > 2 (Stage 1 kidney insufficiency) is [0.587, 0.684]. This is a range of plausible values for the true proportion.