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Paired t test

This analyzes the possibility of an improvement in the quality-of-life measure for ten patients taken at random from those who participated in a trial of a minimally invasive procedure to treat Benign Prostatic Hyperplasia (BPH). BPH is a non-cancerous prostate gland enlargement common to millions of elderly men. The two variables in question are qol_base (quality of life at baseline, before treatment) and qol_3mo (the hypothesized improvement after three months of staying with the treatment.

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Table 1.

Paired Samples Statistics
Mean N Std. Deviation Std. Error Mean
Pair 1 qol_3mo 2.10 10 1.287 .407
qol_base 3.80 10 1.135 .359

Table 1 (alongside) from the output of the paired samples t test shows, first of all, that mean QoL had “fallen” from 3.80 at baseline to 2.10 after three months. This is a positive finding since QoL was a self-rated scale comprising the items: 0=Delighted, 1=Pleased, 2=Mostly satistifed, 3=Mixed, 4=Mostly dissatisfied, 5=Unhappy, and 6=Terrible.

Table 2.

Paired Samples Test
Paired Differences
Mean Std. Deviation Std. Error Mean 95% Confidence Interval of the Difference t df Sig. (2-tailed)
Lower Upper
Pair 1 qol_3mo – qol_base -1.7 1.494 0.473 -2.769 -0.631 -3.597 9 0.006

The derived t value of -3.597 in Table 2 above (ignoring the negative sign owing to the sequence in which the two variables were inserted into the SPSS calculation menu) is so high it bears a significance statistic of p = 0.006. The odds are about 6 in a thousand that such a difference could arise due to chance variation alone. Since this significance result far exceeds the criterion value of α = 0.05, one would have to conclude that BPH patients who received the treatment did experience a subjective improvement.

One-sample t test

Testing the variable delta (delt_qol), which gives the result of the simple calculation qol_3mo minus qol_base, we apply 0 as the test value because the reasonable working hypothesis is that the treatment engenders a perceptible improvement in quality of life ratings.

Table 3.

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One-Sample Statistics
N Mean Std. Deviation Std. Error Mean
delt_qol 10 1.70 1.494 .473

Table 4.

One-Sample Test
Test Value = 0
t df Sig. (2-tailed) Mean Difference 95% Confidence Interval of the Difference
Lower Upper
delt_qol 3.597 9 .006 1.700 .63 2.77

Table 3 (above) shows, first of all, that the mean QoL Delta value is 1.70, suggesting that the 10 patients typically assigned satisfaction ratings two levels higher after three months on the treatment. Table 4 answers the next question, which is that the derived t value of 3.597 for all delta’s minus the benchmark of zero bears a significance statistic of p = 0.006. Once again, the chance that such a large difference from zero could have occurred due to random variation alone is less than six in a thousand. We conclude yet again that the treatment makes a difference for perceived quality of life.

Are Distributions Normal?

Visual inspection of the histograms alone shows that the QoL ratings at baseline roughly approximates a normal distribution while, three months later, the distribution is decidedly skewed toward the “I’m pleased” end of the scale.

Figure 1. Histogram for QoL at Baseline.

Figure 2: Histogram for QoL at Three Months.

Non-parametric Test of Median

If you have a concern about the small sample size and perhaps non-normal data, choose an appropriate nonparametric test to compare the median QoL score at the baseline and at 3 months. (Hint: You need to research nonparametric tests.)

We employ the nonparametric Wilcoxon Rank-Sum Test to compare the medians of what are nominally two independent groups but really the same group of patients answering rating scales at two points in time.

Since we are testing for a leftward shift in QoL ratings at three months and since the mean rank for all negative deltas outnumber the positive ones (Table 5), the correct usage of the p value reported in Table 6 below is to halve it, thus: p = 0.014/2 = 0.007. As it happens, this is lower than the criterion α = 05.

Table 5.

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N Mean Rank Sum of Ranks
qol_3mo – qol_base Negative Ranks 8a 5.38 43.00
Positive Ranks 1b 2.00 2.00
Ties 1c
Total 10
a. qol_3mo < qol_base b. qol_3mo > qol_base
c. qol_3mo = qol_base

Table 6.

Test Statisticsb
qol_3mo – qol_base
Z -2.448a
Asymp. Sig. (2-tailed) .014
a. Based on positive ranks.
b. Wilcoxon Signed Ranks Test

Even switching to non-parametric statistics yields credence to the same conclusion: BPH patients are subjectively better off after three months under the treatment regiment.

Independent Sample Design
Comparing Boxplots
Figure 3: Boxplot of Bone Loss by Case-Control Group.
Comparing Histograms
Figure 4: Comparing Histograms Across Case and Control Groups.
Analysis of Bone Loss Employing a t test

This discretionary test employed the scale variable “change in bone mineral’ density or content and the nominal variable breastfeeding status. Subjects of the study were women who breastfed their newborns and were checked up on at three months of lactation. The control group consisted of formula-feeding mothers and non-pregnant, non- lactating women. Breastfeeding status was used as the grouping variable in the independent-samples t test.

One therefore formulates the hypotheses to be tested as:

H0, the null hypothesis = Among adult women, breastfeeding has no effect on calcium mineral density in bones.

Ha, the alternative hypothesis = Breastfeeding has an effect on bone mineral density in adult women.

Table 7.

Group Statistics
Ref = Laskey N Mean Std. Deviation Std. Error Mean
Percent bone loss control 22 .309 1.2983 .2768
breast-feeding 47 -3.587 2.5056 .3655

Table 8.

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Independent Samples Test
Levene’s Test for Equality of Variances t-test for Equality of Means
F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference
Lower Upper
Percent bone loss Equal variances assumed 11.255 0.001 6.857 67 0.000 3.896 0.568 2.762 5.031
Equal variances not assumed 8.499 66.197 0.000 3.896 0.458 2.981 4.812

Mothers who breastfeed do experience mineral bone loss, as shown by the fact that control women showed an improvement averaging 0.309 on whatever calcium intake they followed (it is not the purpose of the current analysis to account for dietrary and other modifying factors in the Laskey et al. study. The comparable figure for breastfeeding mothers was frank bone mineral loss of 3.59 over the three month run of the study (see Table 7 above). At 3.90, the mean difference in bone mineral change between the two groups lies within the bounds of the 95% confidence interval of the difference (2.8 at the lower bound and 5.0 at the upper limit(Table 8).

A rather high F value for Levene’s Test (11.25, see Table 4) and a test for significance p < 0.05 confirms that the variances in the case-control subject database are homogenous and that one basic assumption of the t test therefore holds. Hence, one scrutinizes the row for "Equal variances assumed". In either case, the derived t score is high enough for the given degrees of freedom to yield a significance statistic that is below the even more rigorous cut-off of p ≤ 0.01. Since such a difference could have occurred purely by chance less than once in a thousand re-sampling passes, one rejects the null hypothesis, accepts the alternate and concludes that breastfeeding has a deleterious effect on bone mineral density. 95% C.I. for Mean Difference in Percentage of Bone Loss The 95% confidence interval is shown below. Table 9. 95% Confidence Interval of the Difference Lower Upper 2.762 5.031 2.981 4.812 Cross-Tabulation Association Between Esophageal Cancer and Alcohol Consumption This is a test of the relationship between alcohol consumption, measured in grams per day, and the nominal variable with two levels ("no" and "yes"), developing esophageal cancer. The latter was used as the grouping variable in the independent-samples t test. The hypotheses may therefore be articulated as follows: One therefore formulates the hypotheses to be tested as: H0, the null hypothesis = there is no linear relationship at all between alcohol consumption levels and esophageal cancer morbidity. Elevated alcohol consumption is not a predictor of developing esophageal cancer. Ha, the alternative hypothesis = Adults presenting with a history of elevated alcohol consumption are at greater risk for esophageal cancer. An exploratory cross-tabulation (Table 10) suggests that those who have not yet developed esophageal cancer are much less likely to be moderate or heavy alcohol drinkers. Nearly half of the control group of 775 adults reported consuming 39 grams a day or less. Nearly all of the remainder, a cumulative 86%, had self-reported alcohol consumption levels of under 80 grams a day. On the other hand, the 200 men who did develop esophageal cancer tended to cluster around 40 to 100 grams daily. Even more revealing, the distribution skewed towards high consumption. Table 10. Alcohol consumption * Esophageal cancer Crosstabulation Esophageal cancer case control Total Alcohol consumption 0 – 39 gm/day Count 29 386 415 % within Esophageal cancer 14.5% 49.8% 42.6% 40 – 79 gm/day Count 75 280 355 % within Esophageal cancer 37.5% 36.1% 36.4% 80 – 119 gm/day Count 51 87 138 % within Esophageal cancer 25.5% 11.2% 14.2% 120+ gm/day Count 45 22 67 % within Esophageal cancer 22.5% 2.8% 6.9% Total Count 200 775 975 % within Esophageal cancer 100.0% 100.0% 100.0% Both the chi-square and odds ratio tests (Table 11) bore significance statistics with p < 0.001, suggesting that the differences between the case and control groups cannot be accounted for by chance variation. Table 11. Chi-Square Tests Value df Asymp. Sig. (2-sided) Pearson Chi-Square 1.590E2 3 .000 Likelihood Ratio 146.498 3 .000 Linear-by-Linear Association 152.974 1 .000 N of Valid Cases 975 a. 0 cells (.0%) have expected count less than 5. The minimum expected count is 13.74. Table 12. Group Statistics Esophageal cancer N Mean Std. Deviation Std. Error Mean Alcohol consumption case 200 2.56 .996 .070 control 775 1.67 .785 .028 This is bolstered by the outcome of the independent samples t test (Table 13, below) where, irrespective of the assumption taken about homogeneity of variances across the case and control groups, the t value is so large that, the significance statistic p < 0.001 shows, the difference in alcohol consumption between case and control groups is extremely unlikely to have occurred due to chance alone. Table 13. Independent Samples Test Levene's Test for Equality of Variances t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper Alcohol consumption Equal variances assumed 36.596 0.000 13.464 973 0.000 0.889 0.066 0.759 1.019 Equal variances not assumed 11.722 266.224 0.000 0.889 0.076 0.740 1.038 Accordingly, one rejects the null hypothesis and concludes that there is a tangible difference in the propensity of esophageal cancer patients to imbibe more alcohol. Applicability of Fisher's Exact Test Fisher's exact test of independence across, in this case, the case and control groups of subjects, applies for 2 X 2 contingency tables. This means Fisher's can be used if the alcohol consumption is recoded as a categorical variable. One would have had to go back to the raw data to reclassify all subjects as either "non-drinker" or "drinker." Alternatively, a better real-world representation might be "non-drinker or light drinker" versus "moderate or heavy drinker." Implementing Fisher's exact test, the null hypothesis would be articulated as that the relative proportions of the dependent variable, having esophageal cancer, are independent of a second variable. So, the null hypothesis would be articulated around the premise that the proportion who drink (or indulge heavily) is equal among those who contract esophageal cancer and perfectly healthy adults. Another consideration that voids the applicability of Fisher's exact test to this data set is that the algorithm is more accurate than the chi-square or the G tests of independence principally when sample sizes are small. In this case, however, the study covered 200 adults in varying stages of esophageal cancer and no les that 775 control subjects. Hence, Fisher's is not a sound choice in this situation. Association Between Esophageal Cancer and Alcohol Consumption as Nominal Variable In fact the database does have a dichotomized version of the alcoholic consumption variable employing 80 grams or more daily as a cut-off for marked indulgence in the vice. The chi-square test is available for testing differences in proportions between categorical variables. We now test for differences in proportions of those who drink relatively little (if at all) and the heavier drinkers between adults who presented with esophageal cancer or did not. The hypotheses may be articulated as follows: H0, the null hypothesis = there is no association between drinking alcohol and esophageal cancer morbidity; put another way, there is no difference in esophageal cancer risk between adults who drink heavily or not. Ha, the alternative hypothesis = Drinking rate and esophageal cancer are related; esophageal cancer risk is greater for adults who are regular or heavy drinkers. Table 14. Esophageal cancer * Alcohol dichotomized Crosstabulation Alcohol dichotomized 80+ gms/day 0-79 gms/day Total Esophageal cancer case Count 96 104 200 % within Alcohol dichotomized 46.8% 13.5% 20.5% control Count 109 666 775 % within Alcohol dichotomized 53.2% 86.5% 79.5% Total Count 205 770 975 % within Alcohol dichotomized 100.0% 100.0% 100.0% Crosstabulation reveals that the likelihood of presenting with esophageal cancer is greater when one is a regular or heavy drinker. Just 13.5 % (or one in seven) of light or non-drinkers "came down" with esophageal cancer. With regular or heavy drinking, the risk rose to just under half. Table 15. Alcohol dichotomized Observed N Expected N Residual 80+ gms/day 205 487.5 -282.5 0-79 gms/day 770 487.5 282.5 Total 975 Table 16. Esophageal cancer Observed N Expected N Residual case 200 487.5 -287.5 control 775 487.5 287.5 Total 975 If the proportions of dichotomized drinking rates are representative of the universe of adults, then around eight in ten are light or non-drinkers. Table 17. Test Statistics Alcohol dichotomized Esophageal cancer Chi-Square 327.410a 339.103a df 1 1 Asymp. Sig. .000 .000 a. 0 cells (.0%) have expected frequencies less than 5. The minimum expected cell frequency is 487.5. Since the computed chi square statistic of 327.4 and 339.1 for the fact of regular alcohol consumption and esophageal cancer morbidity, respectively, (Table 17) exceeds the criterion chi-square statistic of 3.84 at 1 degree of freedom and α = 0.05, we reject the null hypothesis. As well, the probability of a significant difference like this occurring purely by chance is shown to be less than one in a thousand (p < 0.001). Accordingly, we accept the alternative hypothesis and conclude that adults who drink regularly or heavily bear a higher risk of manifesting esophageal cancer sooner or later. Manual Calculation of 95% C.I. of Odds Ratio Estimate The odds ratio can be defined as the ratio of the odds of an event occurring in one group to the odds of it occurring in another group, or to a sample-based estimate of that ratio. These groups might be men and women, an experimental group and a control group, or any other dichotomous classification. If the probabilities of the event in each of the groups are p1 (first group) and p2 (second group), then the odds ratio is: where qx=1-px. An odds ratio of 1 indicates that the condition or event under study is equally likely in both groups. An odds ratio greater than 1 indicates that the condition or event is more likely in the first group. And an odds ratio less than 1 indicates that the condition or event is less likely in the first group. The odds ratio must be greater than or equal to zero if it is defined. It is undefined if p2q1 equals zero.

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