In a study that compared two different medications for the treatment of allergic rhinitis, each patient was asked to rank his or her severity of symptoms on a scale of 0 to 4 (0= none, 1= mild, 2 = moderate, 3= moderately severe, and 4 = severe) 2 hours after the medication was given

6. What type of data will be obtained from this measurement?

7. Which one of the following is appropriate method of central tendency and dispersion for the data that will be obtained above?

A. Mean and standard deviation

B. Median and interquartile range

C. Median and standard deviation

D. Median, no measure of dispersion is acceptable

8. Examine the data set of eight students with the following grades on a possible 100-point examination: 85, 79, 30, 94, 97, 35, 87, and 88. Which one of the following measures of central tendency is least affected by outliers?

A. Median

B. Mean

C. Standard deviation

D. Standard error to the mean

9. A sample of 6 body weights (in pounds) as follows: 116, 168, 124, 132, 110, and 120. The sample median is:

10. A study of 40 patients was performed to examine the association of weight and serum drug levels of a new antibiotic. Linear regression was used to assess the association. The slope of the regression line (slope = 30) was significantly greater than 0, indicating the serum drug level increases as weight increases. The r² value was calculated as 0.75(r=0.86). Which statement provides the most accurate interpretation of these data?

A. Seventy-five percent of the variance in serum levels is likely to be examined by its relationship with weight

B. Twenty-five percent of the variance in serum levels is likely to be explained by its relationship with weight

C . Thirty percent to the variance in serum levels is likely to be explained by its relationship with weight

D. Seventy percent of the variance in serum levels is likely to be explained by its relationship with weight.

tbbrevtatlone investigators use a one-tailed test1 they need to justify its use. There are two ways in which the type I error can be distributed. In a two-tailed test, the rejection region is equally divided between the two ends of the sampling distribution. A sampling distribution can be dened as the relative frequency distribution that would be obtained if all possible samples of a particular sample size were taken. A twortailed test divides the alpha level of 0.05 into both tails. In contrast, a one-tailed test is a test of hypotheSis in which the rejection region is placed entirely at one end of the sampling distribution. A onetailed test puts the 5% in only one tail. A two-tailed test requires a greater difference to produce the same level of statistical signicance as a one-tailed test. The two-tailed test is more conservative and thus preferred in most circumstances. The Slgntcence at No Signicant Difference The failure to nd a difference between (among) a set of data does not necessarily mean that a difference does not exist. Differences may not be detected because of issues with power. Power is the ability of a statistical test to reject the null hypothesis when it is truly false and therefore should be rejected. Type I] error [also referred to as beta or ] is dened as not rejecting the null hypothesis when in actuality it is false; that is. to falsely consider that no difference exists between study groups. Power and type [1 error are related in the equation 1 - type 11 error = power. Statistical power is not an arbitrary number of a study. but rather it is controlled by the design of the study. In studies. a desirable power is at least 30%. This means that there is an 80% chance of detecting a difference between two groups if a difference of a given size really exists. Sample size is related to power; the higher the power that is desired by the investigator, the larger the sample size required. If there are insufcient numbers of patients enrolled in a study. a statistically signicant di'erence will not occur. The sample size is the one element that can easily be manipulated to increase the power. When calculating the sample size for a study that compares two means. several elements are used: desired detectable difference, variability of the samples. and the level of statistical signicance (at). The type I error is typically set at 0.05. There is an inverse relationship between type ] and type II errors. If investigators choose to lower the risk of type I error in a study. they increase the risk of type II error. Therefore. the sample size needs to be increased to compensate for this change. Likewise. effect size (minimum clinically relevant difference] is also determined a priest\" (a prior! is a term used to identify a type of knowledge that agrees with reason and is frequently obtained independent of experience}. and is selected based on clinical judgment and previous literature. There are times when a 1% difference is irrelevant. as in the case of a T093 success rate compared to 71% rate for a new antibiotic compared to standard. In contrast1 investigators may be able to defend a difference of 2% in the rate of a fatal myocardial infarction after receiving a new medication compared to standard therapy. A sufcient number of patients need to be recruited so that any The Science and Practice of Phannacotherapy II [88 clinically meaningful differences are also statistically signicant. Given enough study subjects. any true difference among study groups can be detected at a chosen p value. even if the effect size is clinically unimportant. The smaller the effect size that is clinically important, the greater the number of subjects needed to nd a difference if one truly exists. For xed sample sizes, as the effect size increases. the p value decreases. The clinical question is if it would be worthwhile to enroll these additional subjects to attain statistical signicance if the difference between the two groups is not clinically important. Therefore. it is important for investigators to stipulate the minimum effects when planning a study. The variance is also set at the beginning of the study and is generally based on previous literature. If the variance is low. a given sample of a group is more likely to be representative of the population. Therefore. with lower variance. fewer subjects are needed to reect the underlying population accurately and thus fewer patients are needed to demonstrate a signicant difference if one exists. The best way to prevent a type II error from occurring is to perform a sample size calculation before initiation of the study. Selection at Statlctlcal Tact If the incorrect statistical test is used. a misleading or inaccurate result may occur. There are many statistical tests. and several may be appropriate to use for a given set of data. The test that investigators use needs to be identified in the statistical methods section of the published report and in the footnotes of tables. Several commonly used statistical tests are described in Table 1-]. Among key considerations for choice of an appropriate test is the type of data. whether the data are paired (dependent) or unpaired (independent). and number of groups of data being compared. Statistical tests are also categorized into parametric or nonparametric tests. If appropriate criteria are met. a parametric test is preferred. Parametric tests are used to test differences using interval and ratio data. Samples must be randomly selected from the population and they must be independently measured. In other words. the data should not be paired. matched. correlated. or interdependent in any way. rm variables are independent if knowledge of the value of one variable provides no information about the value of another variable. For example. if you measured blood glucose level and age in a diabetic population. these two variables would in all likelihood be independent. If one knew an individual's blood glucose. this would not provide insight into a person's age. However, the variables were blood glucose and hemoglobin A . then there would be a high degree of dependence. :1 two variables are independent. then the Pearson's correlation {further information on Pearson's correlation is provided in the Regression and Correlation Analysis section) between them is 0. When the phrase \"independence of observation-1+ is used. reference is being made to the concept that if two observations independent of the sampling of one observation do not affect the choice of the second observation. Consider a case in which the observations are not independent. A researcher wants to estimate how productive a person with osteoarthritis is at work compared to others without the disease. The researcher randomly chooses one person who has the t: Phan'nacothcrapy Self-Assessment Program. 4th Edition ibhravlatlona I number of observations (it) was used. In general. the degrees of eedom of an estimate are equal to the number of independent scores that go into the estimate minus the number of parameters estimated. If the average squared deviation was divided by n observations. the variance would be underestimated. As the size of the sample of data increases. the effect of dividing by n or ml is negligible. The sample SD. equal to the square root of the variance. is denoted by the letter s as dened by the following formula: 4.4423\" Using this formula. the SD is the square root of 164M or 6.4. From this example. one can see that each deviation contributes to the SD. Thus. a sample of the same size with less dispersion will have a smaller SD. For example. if the data were changed to: 55. 52. 53. 55. and 50, the mean is the same. but the SD is smaller because the observations lie closer to the mean. The usealness of the SD is related only to normally distributed data. If one assumes that data are normally distributed. then one can say that one SD below and above the mean includes approximately 63% of the observations. two 305 above and below the mean include approximately 95% of the observations. and three SDs in either direction include approximately 99% of the observations. The histogram in Figure l-l describes the distribution of test scores for a larger sample. In Figure l-l. the mean was calculated to be 3?.1'3 and SD is 13.15. Therefore. approximately 68% of the values will be between 24.53 and 50.93 {mean a 1 SD). approximately 95% of individuals will have scores between [1.41% and 54.08 (mean i 2 SD). and approximately 99% of the sample will be between 0 and 31.23 (mean a 3 SD]. The SD and standard mm of the mean {SEMJ are frequently conised terms. The SD indicates the variability of the data around the mean for a sample. whereas the SEM is a measure of precision for the estimated population mean. This estimate is most commonly expressed using a condence interval (Cl) and is related to SD by the equation: SEM = SDhl. The use of' Cls is important in hypothesis testing and is described later in the chapter. \"'1 099099 Testers-ire "Rah-m -I. '30 1 2 3 It 5 s T B El Numberei'lndlvtduah Figure 1-l. Test scores. Pharmacotherapy Self-Assessment Program, 4th Edition 18'? Hypothesis Testing and Meaning at P A hypothesis is an improved theory. The null hypothesis is dened as the theory that no difference exists between study groups. If a study were to compare two means. the null hypothesis {H0} is 11A = lie (i.e.. the population mean of group A is equal to the population mean of group B). The alternate (or research] hypothesis is the theory that a difference does exist between groups. This may be that the mean of group A is greater or less than the mean of group B (ll-a 3* ill; or up. 30). Mean, Medlen, and blade There are three generally accepted measures of central tendency (also referred to as location): the mean, median, and mode. The mean [denoted by f] is one acceptable measure of central tendency for interval and ratio data [Table 1-1]. It is dened by the summation of all values [denoted by X for each data point) divided by the number of subjects in the sample (n) and can be described by the equation Y = Exfn. For instance, the mean number of seizures during a 24-hour period in seven patients with the following values 9, 3, 9, 7, 3, 2, 5, is calculated by dividing the sum of43 by 3', which is equal to 6.14, or an average ofapproximately six seizures per patient. The median is the value where half of the data points fall above and half below it. It is also referred to as the 50th percentile. It is an appropriate measure of central tendency for interval, ratio, and ordinal data. When calculating the median, the rst step is to put the values in rank order. For example, the median nLu'nber of seizures dtuing a 24-hour period in seven patients with the following values 2, 3, S, 'i", 8, 9', 9 is 7. There are three values below and three values above the number 7. If we added one more value [e.g., 1]}, the median would be calculated by taking the two middle numbers and dividing by two. Under these circumstances, the calculation would change to [T + 3);? to get a median of 7.5. Half of the numbers are below 7.5 and half are above. The median has an advantage over mean in that it is affected less by outliers in the data. An outlier is a data point that is an extreme value either much loWer or higher than the rest of the values in the data set. Mathematically, outliers can be determined by using the following formulas: values greater than 1.5 times the interquartile range (IR) plus the upper Biostatistics for the Clinician lhbravlatlons I The ANOVA is preferred over using multiple ttests because when more than one hypothesis is tested on the same data, the risk is greater of making a type I error. If three groups of data were being compared {i.e.., \"A = [.13 = pf), and a Student's ttest was used to compare the means of A versus 13. A versus C, and E versus C', then the type I error rate would be three comparisons times 0.05 or 0.15. If multiple testing did occur. the investigator needs to either use a stricter criterion for signicance or would need to apply the Bonferroni's correction. This factor reduces the threshold p value by the number of comparisons made. For example. if there were six comparisons using multiple ttests, the results would only be accepted as being statistically signicant if the new p value was less than 0.008 rather than 0.05. Twoway {repeated measures) ANOVA is an expansion of the paired t-test and is used when there are more than two groups of data and the same group of subjects is studied using various treatments or time periods. Several assumptions need to be met to use the two-way ANOVA, including independent groups. normally distributed data. similar variance within the groups. and continuous data. The difference between a one-way and two-way ANDVA is that when using a one-way ANOVA there is a single explanatory variable. and a twoway analysis is applied to 2 (two) explanatory variables. The KruskalWallis {oneway] ANOVA is a nonparametric alternative to the oneway ANOVA. The Friedman twoway ANOVA is used as a nonparametric alternative to the two-way ANOVA. For both of these tests. data need to be measured on at least an ordinal scale. Flndlng a Difference wlth Propertlens When a re5earcher has nominal data and want to determine if frequencies are signicantly different from each other for two or more groups. this can be determined by calculating a chi square statistic {X2}. The chi square analysis is one of the most frequently used statistical tests. and compares what is observed with the data with what one would expect to observe if the two variables were independent. If the difference is large enough. researchers conclude that it is statistically signicant. To perfonn a chi square analysis. one must be sure that the data in the contingency table meet several requirements. When using a 23(2 contingency table, ifn is greater than 20. the chi square analysis may be used if all expected frequencies are ve or more. If greater than 2 {two} groups are compared. the chi square may be used if no more than 20% of the cells may have expected 'equencies less than 5 and none may have expected frequenciES less than ]. An example of how to set up a contingency table is as presented in Figure 1-2. A contingency table has two variables. The categories (or levels) of the intervention. the fictitious medication magnadrug or no magnadrug. are represented in k rows in the table and the category of the outcome. gastrointestinal upset, are represented by the m columns in the table. This is a 2X2 contingency table and has 4 (four) cells. A 2X2 contingency table is called this because it has two rows and two columns and "contingency" beeause the values in the cells are contingent on what is happening at the margins. The Science and Practice of Pharmacotherapy II 190 By inspecting the observed frequencies {cells A to D). or those found as a result of the experimental program. there appears to be differences in the numbers of patients who had gastrointestinal upset in each group. The cell frequencies are added to obtain totals for each row and column. An expected frequency is the number of patients one would expect to nd in a given cell if there were no group differences. The formula to calculate the expected frequency ofa cell is as follows: Expected frequency of cell = (cell's row total)(cell's column total) (total number of patients in study) Expected frequency of cell A = {Math}! 6080 (230) = 230 =2\" In the example, all of the expected li'equencies for cells A through D were greater than S. The next step is to detenuine if the frequencies observed in the experiment are signicantly different from the frequencies that would be expected if there were no group di'erences. The chi square statistic is calculated. If the chi square statistic is equal to or greater than the critical value. the difference is considered to be statistically signicant. Chi square analysis does not tell which of the observed differences is statistically signicant from the others. unless there are only two categories of the variable being compared. Further statistical analysis is required to single out specic differences. In this case, n is greater than 20. Ifthe n was lass thanl and if each cell had an expected frequency of at least 5. a Fisher's exact test could have also been used. if more than 2 (two) groups are being compared. a Fisher's exact test may be used if the sample size is at least 20 and any cell has an expected frequency of less than S. The McNemar's test and Cochran's Q test are tests of proportions based on samples that are related. McNemarls test involves dichotomous measurements {e.g.. present or absent) that are paired. Cochran's Q test can be thought of as an extension of the McNemar's test concerned with three or more levels of data. Ragresslon and Correlation Analysis Both regression and correlation analysis are used to evaluate interval or ratio data. Correlation analysis is concerned with determining ifa relationship exists between Gastrointestinal Upset Yes No Total Magnadrug Yes 76 No 204 Total 2 80 Figure l-2. Example ot'a contingency table. Pharrnacotherapy Self-Assessment Program. 4th Edition thbrevletlone condition from an osteoarthritis disease registry and interviews that person. The researcher asks the person who was just interviewed for the name of a friend who can be interviewed next as a control [person without osteoarthritis working the same job]. In this scenario. there is likely to be a strong relationship between the levels of productivity of the two individuals. Thus. a sample ofpeople chosen in this way would consist of dependent pieces of information. In other words. the selection of the rst person would have an inuence on the selection of other subjects in the sample. In short, the observations would not be considered to be independent. The data also need to be normally distributed or the sample must be large enough to make that assumption [central limit theorem] and sample variances must be approximately equal {homogeneity of variance}. The assumption of homogeneity of variance is that the variance within each of the populations is equal. As a rule of thumb. if the largest variance divided by the smaller variance is less than two, then homogeneity may be assumed. This is an assumption of analysis of variance {ANOVA}. which works well even though this assumption is violated except in the case where there are unequal numbers of subjects in various groups. If the variances are not homogeneous. they are heterogeneous. If these characteristics are met. a parametric test may be used. The parametric procedures include tests such as the t-tests. ANOVA. correlation and regression. The list of tests in Table [-1 is not all-inclusive of tests used in clinical trials. but it represents the most corrunon analyses. Complex or uncommon statistical tests may be appropriate. but they should be adequately referenced in the publication of a clinical trial. Comparing Two or More Means The Student's t-test is a parametric statistical test used to test for differences between means of two independent samples. This test was rst described by William Gosset in [903. and was published under the pseudonym \"student". Because the t-test is an example of a parametric test. the criteria for such a test needs to be met before use. The measured variable is approximately normally distributed and continuous. The variances of the two groups are similar. The Student's t-test can be used in cases where there is either an equal or unequal sample size between the two groups. Once the data are collected. and the t value is computed. the researcher consults a table of critical values for t with the appropriate alpha level and degrees of freedom. if the calculated t value is greater than the critical t value. the null hypothesis is rejected and it is concluded that there is a difference between the two groups. In contrast to the Student's Heat. the paired Heat is used in cases in which the same patients are used to collect data for both groups. For example. in a phannacokinetic study where a group of patients have their drug serum concentration measured while taking brand name medication A. and the same group of patients have their drug serum concentration measured while taking medication B. the differences between these two means will Phannacotherapy Self-Assessment Program. 4th Edition 189 be determined using a paired ttest. In this case. patients serve as their own control. With the paired ttest. the tstatistic is not describing differences between the groups. but actual individual patient differences. When the criteria for a parametric test are unable to be met. a nonparametric test can be used. T