Cross-Over Analysis Using T-Tests

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1 Chapter 35 Cross-Over Analysis Using -ests Introuction his proceure analyzes ata from a two-treatment, two-perio (x) cross-over esign. he response is assume to be a continuous ranom variable that follows the normal istribution. In the two-perio cross-over esign, subjects are ranomly assigne to one of two groups. One group receives treatment R followe by treatment. he other group receives treatment followe by treatment R. hus, the response is measure at least twice on each subject. Cross-over esigns are use when the treatments alleviate a conition, rather than effect a cure. After the response to one treatment is measure, the treatment is remove an the subject is allowe to return to a baseline response level. Next, the response to a secon treatment is measure. Hence, each subject is measure twice, once with each treatment. Examples of the situations that might use a cross-over esign are the comparison of anti-inflammatory rugs in arthritis an the comparison of hypotensive agents in essential hypertension. In both of these cases, symptoms are expecte to return to their usual baseline level shortly after the treatment is stoppe. Equivalence Cross-over esigns are popular in the assessment of equivalence. In this case, the effectiveness of a new treatment formulation (rug) is to be compare against the effectiveness of the currently use (reference) formulation. When showing equivalence, it is not necessary to show that the new treatment is better than the current treatment. Rather, the new treatment nee only be shown to be as goo as the reference so that it can be use in its place. Avantages of Cross-Over Designs A comparison of treatments on the same subject is expecte to be more precise. he increase precision often translates into a smaller sample size. Also, patient enrollment into the stuy may be easier because each patient will receive both treatments. Disavantages of Cross-Over Designs he statistical analysis of a cross-over experiment is more complex than a parallel-group experiment an requires aitional assumptions. It may be ifficult to separate the treatment effect from the time effect an the carry-over effect of the previous treatment. he esign cannot be use when the treatment (or the measurement of the response) alters the subject permanently. Hence, it cannot be use to compare treatments that are intene to effect a cure. 35-

2 Because subjects must be measure at least twice, it may be more ifficult to keep patients enrolle in the stuy. It is arguably simpler to measure a subject once than to obtain their measurement twice. his is particularly true when the measurement process is painful, uncomfortable, embarrassing, or time consuming. echnical Details Cross-Over Analysis In the iscussion that follows, we summarize the presentation of Chow an Liu (999). We suggest that you review their book for a more etaile presentation. he general linear moel for the stanar x cross-over esign is Y + S + P + F + C + e ijk µ ik j j, k j, k ijk ( ) ( ) where i represents a subject ( to n k ), j represents the perio ( or ), an k represents the sequence ( or ). he S ik represent the ranom effects of the subjects. he P j represent the effects of the two perios. he F( j, k ) represent the effects of the two formulations (treatments). In the case of the x cross-over esign F ( j k ), FR if k j F if k j where the subscripts R an represent the reference an treatment formulations, respectively. C j he (, k ) represent the carry-over effects. In the case of the x cross-over esign C ( j, k ) CR if j, k C if j, k 0 otherwise where the subscripts R an represent the reference an treatment formulations, respectively. Assuming that the average effect of the subjects is zero, the four means from the x cross-over esign can be summarize using the following table. where P + P, F + F 0, an C + C 0. 0 R Sequence Perio Perio ( R ) µ µ + P + F µ µ + P + F + C ( R) µ µ + P + F µ µ + P + F + C R R R R Carryover Effect he x cross-over esign shoul only be use when there is no carryover effect from one perio to the next. he presence of a carryover effect can be stuie by testing whether C CR 0 using a t test. his test is calculate as follows c σ u C + 35-

3 where C U U U k n k k i U n ik n σ u ik k k ( ) n n k i U Y + Y ik ik i k ( U U ) he null hypothesis of no carryover effect is rejecte at the α significance level if > tα /, +. c A 00( α )% confience interval for C C C is given by R ( tα /, + ) C ± σ u +. reatment Effect he presence of a treatment (rug) effect can be stuie by testing whether F calculate as follows where σ F + F 0 using a t test. his test is R F k n k k i n ik n σ ik k k ( ) n n k i ik Y Y ik ik ( ) he null hypothesis of no rug effect is rejecte at the α significance level if > tα /, + A 00( α )% confience interval for F F F is given by R ( tα /, + ) F ± σ

4 Perio Effect he presence of a perio effect can be stuie by testing whether P P 0 using a t test. his test is calculate as follows where P σ P + P O O O O n σ ik k k ( ) n n k i ik Y Y ik ik ( ) he null hypothesis of no rug effect is rejecte at the α significance level if > tα /, +. P A 00( α )% confience interval for P P P is given by ( tα /, + ) P ± σ +. Bioequivalence he t test of formulations (treatments) may be thought of as a preliminary assessment of bioequivalence. However, this t test investigates whether the two treatments are ifferent. It oes not assess whether the two treatments are the same bioequivalent. hat is, failure to reject the hypothesis of equal means oes not imply bioequivalence. In orer to establish bioequivalence, ifferent statistical tests must be use. Before iscussing these tests, it is important to unerstan that, unlike most statistical hypothesis tests, when testing bioequivalence, you want to establish that the response to the two treatments is the same. Hence, the null hypothesis is that the mean responses are ifferent an the alternative hypothesis is that the mean responses are equal. his is just the opposite from the usual t test. his is why bioequivalence testing requires the special statistical techniques iscusse here. When using a cross-over esign to test for bioequivalence, a washout perio between the first an secon perios must be use that is long enough to eliminate the resiual effects of the first treatment from the response to the secon treatment. Because of this washout perio, there is no carryover effect. Without a carryover effect, the general linear moel reuces to Y + S + P + F + e ijk µ ik j j, k ijk here are many types of bioequivalence. he x cross-over esign is use to assess average bioequivalence. Remember that average bioequivalence is a statement about the population average. It oes not make reference to the variability in responses to the two treatments. he 99 FDA guiance uses the ± 0 rule which allows an ( ) 35-4

5 average response to a test formulation to vary up to 0% from the average response of the reference formulation. his rule requires that ratio of the two averages µ / µ R be between 0.8 an. (80% to 0%). Another way of stating this is that the µ is within 0% of µ R. he FDA requires that the significance level be 0.0 or less. Several methos have been propose to test for bioequivalence. Although the program provies several methos, you shoul select only the one that is most appropriate for your work. Confience Interval Approach he confience interval approach, first suggeste by Westlake (98), states that bioequivalence may be conclue if a ( α ) 00% confience interval for the ifference µ µ R or ratio µ / µ R is within acceptance limits (α is usually set to 0.05). If the ± 0 rule is use, this means that the confience interval for the ifference must be between -0. an 0.. Likewise, the confience interval for the ratio must be between 0.8 an. (or 80% an 0%). Several methos have been suggeste for computing the above confience interval. he program provies the results for five of these. Perhaps the best of the five is the one base on Fieller s heorem since it makes the fewest, an most general, assumptions about the istribution of the responses. Classic (Shortest) Confience Interval of the Difference L ( Y YR ) ( t ) α, + σ + ( ) ( α ) + U Y YR + t, σ + Classic (Shortest) Confience Interval of the Ratio A confience interval for the ratio may be calculate from the confience interval on the ifference using the formula ( / ) ( / ) L L Y R + 00% U U Y R + 00% Westlake s Symmetric Confience Interval of the Difference First, compute values of k an k so that Next, compute using k α n + n t k σ + k ( Y Y ) R k σ + + ( Y Y ) R Finally, conclue bioequivalence if < 0. µ R 35-5

6 Westlake s Symmetric Confience Interval of the Ratio A confience interval for the ratio may be calculate from the confience interval on the ifference using the formula ( / Y R ) L % ( / Y R ) U % Confience Interval of the Ratio Base on Fieller s heorem Both the classic an Westlake s confience interval for the ratio o not take into account the variability of Y R an the correlation between Y R an Y Y. Locke (984) provies formulas using Fieller s theorem that oes take R into account the variability of Y R. his confience interval is popular not only because it takes into account the variability of Y R, but also the intersubject variability. Also, it only assumes that the ata are normal, but not that the group variances are equal as o the other two approaches. R he ( α ) 00% confience limits for δ µ µ where S R / are the roots of the quaratic equation ( Y δyr ) ( tα, n + n ) ω( S δsr δ S RR) S S RR + 0 ( n n ) ( n n ) ( n n ) n ω + 4 n ( Yi Y ) + ( Yi Y ) i n n i ( Yi Y ) + ( Yi Y ) i n i ( Yi Y )( Yi Y ) + ( Yi Y )( Yi Y ) i Aitionally, in orer for the roots of the quaratic equation to be finite positive real numbers, the above values must obey the conitions an YR ωs Y ωs RR > t + α, n n > tα n + n, n i Interval Hypotheses esting Approach Schuirmann (98) introuce the iea of using an interval hypothesis to test for average bioequivalence using the following null an alternative hypotheses H0: µ µ R θ L or µ µ R θu 35-6

7 H a :θ L < µ µ R < θ U where θ L an θ U are limits selecte to insure bioequivalence. Often these limits are set at 0% of the reference mean. hese hypotheses can be rearrange into two one-sie hypotheses as follows H : µ µ θ versus H : µ µ > θ 0 R L a R L H : µ µ θ versus H : µ µ < θ 0 R U a R U he first hypothesis test whether the treatment response is too low an the secon tests whether the treatment response is too high. If both null hypotheses are rejecte, you conclue that the treatment rug is bioequivalent to the reference rug. Schuirmann s wo One-Sie ests Proceure Schuirmann s proceure is to conuct two one-sie tests, each at a significance level of α. If both tests are rejecte, the conclusion of bioequivalence is mae at the α significance level. hat is, you conclue that µ an µ R are average equivalent at the α significance level if an U L ( Y Y ) σ θ + R L ( Y Y ) σ θ + R U t > α, n + n > tα, + n n Wilcoxon-Mann-Whitney wo One-Sie ests Proceure When the normality assumption is suspect, you can use the nonparametric version of Schuirmann s proceure, known as the Wilcoxon-Mann-Whitney two one-sie tests proceure. his rather complicate proceure is escribe on pages 0-5 of Chow an Liu (999) an we will not repeat their presentation here. Anerson an Hauck s est Unlike Schuirman s test, Anerson an Hauck (983) propose a single proceure that evaluates the null hypothesis of inequivalence versus the alternative hypothesis of equivalence. he significance level of the Anerson an Hauck test is given by where ( tah ) ( tah ) α Pr δ Pr δ ( ) x n + n Pr x t t δ θu θl σ

8 t AH ( Y YR ) ( θu + θl) σ + / Data Structure he ata for a cross-over esign is entere into three variables. he first variable contains the sequence number, the secon variable contains the response in the first perio, an the thir variable contains the response in the secon perio. Note that each row of ata represents the complete response for a single subject. Chow an Liu (999) give the following ata on page 73. We will use these ata in our examples to verify the accuracy of our calculations. hese ata are containe in the ataset calle ChowLiu73. ChowLiu73 ataset Sequence Perio Perio Valiation Chow an Liu (999) use the above ataset throughout their book. Except for some obvious typographical errors that exist in their book, our results match their results exactly. We have also teste the algorithm against examples in other texts. In all cases, NCSS matches the publishe results. 35-8

9 Proceure Options his section escribes the options available in this proceure. o fin out more about using a proceure, turn to the Proceures chapter. Following is a list of the proceure s options. Variables ab he options on this panel specify which variables to use. Sequence Variable Sequence Group Variable Specify the variable containing the sequence number. he values in this column shoul be either (for the first sequence) or (for the secon sequence). In the case of a bioequivalence stuy, the program assumes that the reference rug is aministere first in sequence an secon in sequence. Perio Variables Perio Variable Specify the variable containing the responses for the first perio of the cross-over trial, one subject per row. Perio Variable Specify the variable containing the responses for the secon perio of the cross-over trial, one subject per row. reatment Labels Label his is the one-letter label given to the first treatment. his ientifies the treatment that occurs first in sequence. In an equivalence trial, this is the label of the reference formulation. Common choices are R or A. Label his is the one-letter label given to the secon treatment. his ientifies the treatment that occurs secon in sequence. In an equivalence trial, this is the label of the treatment formulation. Common choices are or B. Alpha Levels Cross-Over Alpha Level his is the value of alpha use in the cross-over reports. One minus alpha is the confience level of the confience intervals in the cross-over reports. For example, setting alpha to 0.05 results in a 95% confience interval. A value of 0.05 is commonly use. For the preliminary tests, using 0.0 is common. You shoul not be afrai to use other values since 0.05 became popular in pre-computer ays when it was the only value available. ypical values range from 0.00 to 0.0. Equivalence Alpha Level his is the value of alpha use in the equivalence reports. One minus alpha is the confience level of the confience intervals in the equivalence reports. For example, setting alpha to 0.05 results in a 95% confience interval. 35-9

10 You shoul not be afrai to use values other than 0.05 since this value became popular in pre-computer ays when it was the only value available. ypical values range from 0.00 to 0.0. Equivalence Limits Upper Equivalence Limit Specify the upper limit of the range of equivalence. Differences between the two treatment means greater than this amount are consiere to be bioinequivalent. Note that this shoul be a positive number. If the % box is checke, this value is assume to be a percentage of the reference mean. If the % box is not checke, this value is assume to be the value of the ifference. Lower Equivalence Limit Specify the lower limit of the range of equivalence. Differences between the two treatment means less than this amount are consiere to be bioinequivalent. Note that this shoul be a negative number. If the % box is checke, this value is assume to be a percentage of the reference mean. If the % box is not checke, this value is assume to be the value of the ifference. If you want symmetric limits, enter -UPPER LIMI here an the negative of the Upper Equivalence Limit will be use. Reports ab he options on this panel control the reports an plots. Select Reports Cross-Over Summary Report Written Explanations Each of these options inicates whether to isplay the inicate reports. Written Explanations Inicate whether to isplay the written explanations an interpretations that can be isplaye following each report an plot. Report Options Variable Names his option lets you select whether to isplay only variable names, variable labels, or both. Value Labels his option applies to the Group Variable(s). It lets you select whether to isplay ata values, value labels, or both. Use this option if you want the output to automatically attach labels to the values (like Yes, No, etc.). See the section on specifying Value Labels elsewhere in this manual. Precision Specify the precision of numbers in the report. A single-precisioumber will show seven-place accuracy, while a ouble-precisioumber will show thirteen-place accuracy. Note that the reports were formatte for single precision. If you select ouble precision, some numbers may run into others. Also note that all calculations are performe in ouble precision regarless of which option you select here. his is for reporting purposes only. 35-0

11 Report Options Decimal Places Mean... est Decimals Specify the number of igits after the ecimal point to isplay on the output of values of this type. Note that this option io way influences the accuracy with which the calculations are one. Plots ab he options on this panel control the appearance of various plots. Select Plots Means Plot Probability Plots Each of these options inicates whether to isplay the inicate plots. Click the plot format button to change the plot settings. Example Cross-Over Analysis an Valiation his section presents an example of how to run an analysis of ata from a x cross-over esign. Chow an Liu (999) page 73 provie an example of ata from a x cross-over esign. hese ata were shown in the Data Structure section earlier in this chapter. On page 77, they provie the following summary of the results of their analysis. Effect MVUE Variance 95% CI P-Value Carryover (-4.0,.9) reatment (-0.03, 5.46) Perio (-9.47, 6.0) We will use the ata foun the CHOWLIU73 atabase. You may follow along here by making the appropriate entries or loa the complete template Example from the emplate tab of the Cross-Over Analysis Using - ests winow. Open the ChowLiu73 ataset. From the File menu of the NCSS Data winow, select Open Example Data. Click on the file ChowLiu73.NCSS. Click Open. Open the winow. On the menus, select Analysis, then -ests, then. he Cross-Over Analysis Using -ests proceure will be isplaye. On the menus, select File, then New emplate. his will fill the proceure with the efault template. 3 Specify the variables. On the winow, select the Variables tab. Double-click in the Sequence Group Variable box. his will bring up the variable selection winow. Select Sequence from the list of variables an then click Ok. he phrase Sequence will appear in the Perio Variable box. Double-click in the Perio Variable box. his will bring up the variable selection winow. 35-

12 Select Perio from the list of variables an then click Ok. he phrase Perio will appear in the Perio Variable box. Remember that you coul have entere a here signifying the secon variable on the ataset. Double-click in the Perio Variable box. his will bring up the variable selection winow. Select Perio from the list of variables an then click Ok. he phrase Perio will appear in the Perio Variable box. 4 Run the proceure. From the Run menu, select Run Proceure. Alternatively, just click the green Run button. he following reports an charts will be isplaye in the Output winow. Cross-Over Analysis Summary Section Lower 95.0% Upper 95.0% Estimate Stanar Value Prob Confience Confience Parameter Effect Error (DF) Level Limit Limit reatment Perio Carryover Interpretation of the Above Report he two treatment means in a x cross-over stuy are not significantly ifferent at the significance level (the actual significance level was ). he esign ha subjects in sequence (R) an subjects in sequence (R). he average response to treatment R was 8.56 an the average response to treatment was A preliminary test faile to reject the assumption of equal perio effects at the significance level (the actual significance level was ). A preliminary test faile to reject the assumption of equal carryover effects at the significance level (the actual significance level was ). his report summarizes the results of the analysis. he reatment line presents the results of the t-test of whether the treatments are ifferent. he Perio line presents the results of a preliminary test of the assumption that the perio effects are equal. he Carryover line presents the results of a preliminary test of the assumption that there is no carryover effect. his is a critical assumption. If the carryover effect is significant, you shoul not be using a cross over esign. Note that the values in this report match the values from page 77 of Chow an Liu which valiates this part of the program. Parameter hese are the items being teste. Note that the reatment line is the main focus of the analysis. he Perio an Carryover lines are preliminary tests of assumptions. Estimate Effect hese are the estimate values of the corresponing effects. Formulas for the three effects were given in the echnical Details section earlier in this chapter. Stanar Error hese are the stanar errors of each of the effects. hey provie an estimate of the precision of the effect estimate. he formulas were given earlier in the echnical Details section of this chapter. Value (DFxx) hese are the test statistics calculate from the ata that are use to test whether the effect is ifferent from zero. he DF is the value of the egrees of freeom. his is two less than the total number of subjects in the stuy. 35-

13 Prob Level his is the probability level (p-value) of the test. If this value is less than the chosen significance level, then the corresponing effect is sai to be significant. For example, if you are testing at a significance level of 0.05, then probabilities that are less than 0.05 are statistically significant. You shoul choose a value appropriate for your stuy. Some authors recommen that the tests of assumptions (Perio an Carryover) shoul be one at the 0.0 level of significance. Upper an Lower Confience Limits α 00% confience interval for the estimate effect. hese values provie a ( ) Interpretation of the Above Report his section provies a written interpretation of the above report. Cross-Over Analysis Detail Section Least Squares Stanar Stanar Seq. Perio reatment Count Mean Deviation Error R R Difference (-R)/ Difference (-R)/ otal R otal R R Interpretation of the Above Report his report shows the means an stanar eviations of various subgroups of the ata. he least squares mean of treatment R is 8.56 an of treatment is Note that least squares means are create by taking the simple average of their component means, not by taking the average of the raw ata. For example, if the mean of the 0 subjects in perio sequence is 50.0 an the mean of the 0 subjects in perio sequence is 40.0, the least squares mean is ( )/ hat is, no ajustment is mae for the unequal sample sizes. Also note that the stanar eviation an stanar error of some of the subgroups are not calculate. his report provies the least squares means of various subgroups of the ata. Seq. his is the sequence number of the mean shown on the line. When the ot (perio) appears in this line, the results isplaye are create by taking the simple average of the appropriate means of the two sequences. Perio his is the perio number of the mean shown on the line. When the ot (perio) appears in this line, the results isplaye are create by taking the simple average of the appropriate means of the two perios. reatment his is the treatment (or formulation) of the mean shown on the line. When the ot (perio) appears in this line, the results isplaye are create by taking the simple average of the appropriate means of the two treatments. When the entry is (-R)/, the mean is compute on the quantities create by iviing the ifference in each subject s two scores by. When the entry is R+, the mean is compute on the sums of the subjects two scores. 35-3

14 Count he count is the number of subjects in the mean. Least Squares Mean Least squares means are create by taking the simple average of their component means, not by taking a weighte average base on the sample size in each component. For example, if the mean of the 0 subjects in perio sequence is 50.0 an the mean of the 0 subjects in perio sequence is 40.0, the least squares mean is ( )/ hat is, no ajustment is mae for the unequal sample sizes. Since least squares means are use in all subsequent calculations, these are the means that are reporte. Stanar Deviation his is the estimate stanar eviation of the subjects in the mean. Stanar Error his is the estimate stanar error of the least squares mean. Equivalence Base on the Confience Interval of the Difference Equivalence Base on the Confience Interval of the Difference Lower Lower 90.0% Upper 90.0% Upper Equivalent Equivalence Confience Confience Equivalence at the 5.0% est ype Limit Limit Limit Limit Sign. Level? Shortest C.I Yes Westlake C.I Yes Note: Westlake's k -.37 an k.60. Interpretation of the Above Report Average bioequivalence of the two treatments has been foun at the significance level using the shortest confience interval of the ifference approach since both confience limits, an 4., are between the acceptance limits of -6.5 an 6.5. his experiment use a x cross-over esign with subjects in sequence an subjects in sequence. Average bioequivalence of the two treatments has been foun at the significance level using Westlake's confience interval of the ifference approach since both confience limits, -7.4 an 7.4, are between the acceptance limits of -6.5 an 6.5. his experiment use a x cross-over esign with subjects in sequence an subjects in sequence. his report provies the results of two tests for bioequivalence base on confience limits of the ifference between the means of the two formulations. est ype his is the type of test reporte on this line. he mathematical etails of each test were escribe earlier in the echnical Details section of this chapter. Lower an Upper Equivalence Limit hese are the limits on bioequivalence. As long as the ifference between the treatment formulation an reference formula is insie these limits, the treatment formulation is bioequivalent. hese values were set by you. hey are not calculate from the ata. Lower an Upper Confience Limits hese are the confience limits on the ifference in response to the two formulations compute from the ata. Note that the confience coefficient is ( α ) 00%. If both of these limits are insie the two equivalence limits, the treatment formulation is bioequivalent to the reference formulation. Otherwise, it is not. Equivalent at the 5.0% Sign. Level? his column inicates whether bioequivalence can be conclue. 35-4

15 Equivalence Base on the Confience Interval of the Ratio Lower Lower 90.0% Upper 90.0% Upper Equivalent Equivalence Confience Confience Equivalence at the 5.0% est ype Limit Limit Limit Limit Sign. Level? Shortest C.I Yes Westlake C.I Yes Fieller's C.I Yes Interpretation of the Above Report Average bioequivalence of the two treatments has been foun at the significance level using the shortest confience interval of the ratio approach since both confience limits, an 04.99, are between the acceptance limits of an his experiment use a x cross-over esign with subjects in sequence an subjects in sequence. Average bioequivalence of the two treatments has been foun at the significance level using Westlake's confience interval of the ratio approach since both confience limits, 9.0 an 08.98, are between the acceptance limits of an his experiment use a x cross-over esign with subjects in sequence an subjects in sequence. Average bioequivalence of the two treatments has been foun at the significance level using Fieller's confience interval of the ratio approach since both confience limits, an 04.9, are between the acceptance limits of an his experiment use a x cross-over esign with subjects in sequence an subjects in sequence. his report provies the results of three tests for bioequivalence base on confience limits of the ratio of the mean responses to the two formulations. est ype his is the type of test report on this line. he mathematical etails of each test were escribe earlier in the echnical Details section of this chapter. Lower an Upper Equivalence Limit hese are the limits on bioequivalence in percentage form. As long as the percentage of the treatment formulation of the reference formula is between these limits, the treatment formulation is bioequivalent. hese values were set by you. hey are not calculate from the ata. Lower an Upper Confience Limits hese are the confience limits on the ratio of mean responses to the two formulations compute from the ata. Note that the confience coefficient is ( α ) 00%. If both of these limits are insie the two equivalence limits, the treatment formulation is bioequivalent to the reference formulation. Otherwise, it is not. Equivalent at the 5.0% Sign. Level? his column inicates whether bioequivalence can be conclue. 35-5

16 Equivalence Base on Schuirmann s wo One-Sie Hypothesis ests Lower Upper 5.0% Equivalent est est Cutoff at the 5.0% est ype Value Value Value DF Sign. Level? Schuirmann's -Sie ests Yes Interpretation of the Above Report Average bioequivalence of the two treatments was foun at the significance level using Schuirmann's two one-sie t-tests proceure. he probability level of the t-test of whether the treatment mean is not too much lower than the reference mean is he probability level of the t-test of whether the treatment mean is not too much higher than the reference mean is Since both of these values are less than , the null hypothesis of average bioinequivalence was rejecte in favor of the alternative hypothesis of average bioequivalence. his experiment use a x cross-over esign with subjects in sequence an subjects in sequence. his report provies the results of Schuirmann s two one-sie hypothesis tests proceure. est ype his is the type of test reporte on this line. he mathematical etails of this test were escribe earlier in the echnical Details section of this chapter. Lower an Upper est Value hese are the values of L an U, the two one-sie test statistics. 5% Cutoff Value his is the value that marks significance or non-significance. If the absolute values of both L an U are greater than this value, the treatment formulation is bioequivalent. Otherwise, it is not. his value is base on the egrees of freeom an on α. DF his is the value of the egrees of freeom. In this case, the value of the egrees of freeom is n Equivalent at the 5.0% Sign. Level? his column inicates whether bioequivalence is conclue. + n. Equivalence Base on wo One-Sie Wilcoxon-Mann-Whitney ests Lower Lower Upper Upper Equivalent Sum Prob Sum Prob at the 5.0% est ype Ranks Level Ranks Level Sign. Level? -Sie MW ests Yes Interpretation of the Above Report Average bioequivalence of the two treatments was foun at the significance level using the nonparametric version of Schuirmann's two one-sie tests proceure which is base on the Wilcoxon-Mann-Whitney test. he probability level of the test of whether the treatment mean is not too much lower than the reference mean is he probability level of the test of whether the treatment mean is not too much higher than the reference mean is Since both of these values are less than , the null hypothesis of average bioinequivalence was rejecte in favor of the alternative hypothesis of average bioequivalence. his experiment use a x cross-over esign with subjects in sequence an subjects in sequence. his report provies the results of the nonparametric version of Schuirmann s two one-sie hypothesis tests proceure. 35-6

17 est ype his is the type of test reporte on this line. he mathematical etails of this test were escribe earlier in the echnical Details section of this chapter. Lower an Upper Sum Ranks hese are sum of the ranks for the lower an upper Mann-Whitney tests. Lower an Upper Prob Level hese are the upper an lower significance levels of the two one-sie Wilcoxon-Mann-Whitney tests. Bioequivalence is inicate when both of these values are less than a given level of α. Equivalent at the 5.0% Sign. Level? his column inicates whether bioequivalence is conclue. Equivalence Base on Anerson an Hauck s Hypothesis est Equivalent Prob at the 5.0% est ype Pr(-L) Pr(U) Level Sign. Level? Anerson an Hauck's est Yes Interpretation of the Above Report Average bioequivalence of the two treatments was foun at the significance level using Anerson an Hauck's test proceure. he actual probability level of the test was his experiment use a x cross-over esign with subjects in sequence an subjects in sequence. his report provies the results of Anerson an Hauck s hypothesis test proceure. est ype his is the type of test reporte on this line. he mathematical etails of this test were escribe earlier in the echnical Details section of this chapter. Pr(-L) an Pr(U) hese values are subtracte to obtain the significance level of the test. Prob Level his is the significance level of the test. Bioequivalence is inicate when this value is less than a given level of α. Equivalent at the 5.0% Sign. Level? his column inicates whether bioequivalence is conclue. 35-7

18 Plot of Sequence-by-Perio Means he sequence-by-perio means plot shows the mean responses on the vertical axis an the perios on the horizontal axis. he lines connect like treatments. he istance between these lines represents the magnitue of the treatment effect. If there is no perio, carryover, or interaction effects, two horizontal lines will be isplaye. he tenency for both lines to slope up or own represents perio an carryover effects. he tenency for the lines to cross represents perio-by-treatment interaction. his is also a type of carryover effect. Plot of Subject Profiles he profile plot isplays the raw ata for each subject. he response variable is shown along the vertical axis. he two sequences are shown along the horizontal axis. he ata for each subject is epicte by two points connecte by a line. he subject s response to the reference formulation is shown first followe by their response to the treatment formulation. Hence, for sequence, the results for the first perio are shown on the right an for the secon perio on the left. 35-8

19 his plot is use to evelop a feel for your ata. You shoul view it first as a tool to check for outliers (points an subjects that are very ifferent from the majority). Note that outliers shoul be remove from the analysis only if a reason can be foun for their eletion. Of course, the first step in ealing with outliers is to ouble-check the ata values to etermine if a typing error might have cause them. Also, look for subjects whose lines exhibit a very ifferent pattern from the rest of the subjects in that sequence. hese might be a signal of some type of atarecoring or ata-entry error. he profile plot allows you to assess the consistency of the responses to the two treatments across subjects. You may also be able to evaluate the egree to which the variation is equal in the two sequences. Plot of Sums an Differences he sums an ifferences plot shows the sum of each subject s two responses on the horizontal axis an the ifference between each subject s two responses on the vertical axis. Dot plots of the sums an ifferences have been ae above an to the right, respectively. Each point represents the sum an ifference of a single subject. Different plotting symbols are use to enote the subject s sequence. A horizontal line has been ae at zero to provie an easy reference from which to etermine if a ifference is positive (favors treatment R) or negative (favors treatment ). he egree to which the plotting symbols ten to separate along the horizontal axis represents the size of the carryover effect. he egree to which the plotting symbols ten to separate along the vertical axis represents the size of the treatment effect. Outliers are easily etecte on this plot. Outlying subjects shoul be reviewe for ata-entry errors an for special conitions that might have cause their responses to be unusual. Outliers shoul not be remove from an analysis just because they are ifferent. A compelling reason shoul be foun for their removal an the removal shoul be well ocumente. 35-9

20 Perio Plot he Perio Plot isplays a subject s perio response on the horizontal axis an their perio response on the vertical axis. he plotting symbol is the sequence number. he plot is use to fin outliers an other anomalies. Probability Plots hese plots show the ifferences (P-P) on the vertical axis an values on the horizontal axis that woul be expecte if the ifferences were normally istribute. he first plot shows the ifferences for sequence an the secon plot shows the ifferences for sequence. If the assumption of normality hols, the points shoul fall along a straight line. he egree to which the points are off the line represents the egree to which the normality assumption oes not hol. Since the normality of these ifferences is assume by the t-test use to test for a ifference between the treatments, these plots are useful in assessing whether that assumption is vali. If the plots show a pronounce pattern of non-normality, you might try taking the square roots or the logs of the responses before beginning the analysis. 35-0

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