“MANOVA OVER ANOVA” A BETTER OBJECTIVE IN BIOEQUIVALENCE STUDY
HTML Full Text“MANOVA OVER ANOVA” A BETTER OBJECTIVE IN BIOEQUIVALENCE STUDY
S. Patel*^{1}, H. Padh ^{2} and C. Bhavsar ^{1}
Department of Statistics, University School of Sciences, Gujarat University ^{1}, Navrangpura, Ahmedabad380009, Gujarat, India
Sardar Patel University ^{2}, Gujarat, India
ABSTRACT: Bioequivalence studies should be conducted for two products marketed by different licensees, containing same active ingredient(s), must be shown to be therapeutically equivalent to one another order to be considered interchangeable. The bioequivalence of two formulations of the same drug can be determined based on the absence of significant differences in primary pharmacokinetic properties of bioavailability, such as pharmacokinetic parameters like C_{max}, T_{max}, AUC_{0t, }and AUC_{0∞}. The pharmacokinetic parameters derived from the plasma concentrationtime curve are subjected to ANOVA. So we need to check ANOVAs for all pharmacokinetic parameters. Instead of that we can use multivariate analysis of variance (MANOVA) as it contains ANOVA results and further give more information regarding significance. From the results we can see that we get the same values like ANOVA and additionally we get 4 different tests for significance. Wilk’s Lambda shows that 6.9%, 14.1% and 20% of the variance of the dependent variable (C_{max}, T_{max}, AUC_{0t, }and AUC_{0∞}) is accounted for by the differences between drugs, phase and interaction respectively. Pillai’s Trace, Hotelling’s Trace and Roy’s largest root says that the data lead to statistical insignificance. So from these results we can suggest MANOVA instead of ANOVA in bioequivalence and control the increase risk of Type I error.
Keywords:
Pharmacokinetic, Bioequivalence, Significance, Doubleblind
INTRODUCTION: Bioequivalence studies should be conducted for the comparison of two medicinal products containing the same active substance, also compare the expected in vivo biological equivalence of two formulations of a drug ^{1–4}.
The studies should provide an objective means of critically assessing the possibility of alternative use of them. Two products marketed by different licensees, containing same active ingredient(s), must be shown to be therapeutically equivalent to one another order to be considered interchangeable ^{5}. The bioequivalence of two formulations of the same drug can be determined based on the absence of significant differences in primary pharmacokinetic properties of bioavailability, such as the rates of absorption and elimination, the extent of absorption or total amount of drug absorbed in the body ^{6}.
Pharmacokinetic parameters and Statistical Analysis: C_{max} (This is the maximum drug concentration achieved in systemic circulation following drug administration.), T_{max} (It is the time required to achieve maximum drug concentration in systemic circulation.), AUC_{0t} (Area under the plasma concentrationtime curve from 0 to the last quantifiable concentration to be calculated using the trapezoidal rule.), AUC_{0∞} (Area under the plasma concentrationtime curve, from zero to infinity to be calculated as the sum of AUC_{0t} plus the ratio of the last measurable concentration to the elimination rate constant). Maximal plasma concentration (C_{max}) and time to reach the peak concentration (T_{max}) were obtained directly by the visual inspection of each subject's plasma concentrationtime profile.The AUC_{0t} from time zero to the last quantifiable point (Ct) was calculated using the trapezoidal rule and the extrapolated AUC from Ct to infinity (AUC_{0∞}) was determined as Ct/K_{el}. The area under the plasma concentrationtime from 0 to infinity (AUC_{0∞}) was calculated as the sum of the AUC_{0t} plus the ratio of the last measurable concentration to the elimination rate constant. To test the bioequivalence of the test and reference formulations, analysis of variance (ANOVA) for the crossover design was conducted on logtransformed C_{max}, T_{max,} AUC_{0–t}, and AUC_{0–∞}. The pharmacokinetic parameters derived from the plasma concentrationtime curve are subjected to ANOVA in which the variance is partitioned into components due to subjects, periods and treatments. In ANOVA null hypothesis is of equal means, test and reference are equivalent (i.e. H_{0}: µ_{T} = µ_{R}), where µ_{T }and µ_{R }represents the expected mean bioavailabilities of the test and reference formulations, respectively. The alternate hypothesis is test and reference is bioinequivalent. (i.e. H_{0}: µ_{T} ≠ µ_{R}). For a crossover trial with n subjects and t treatments, the ANOVA takes the form as shown in Table 1 ^{3}.
TABLE 1: ANALYSIS OF VARIANCE (ANOVA) TABLE FOR tPERIOD, TTREATMENT CROSSOVER DESIGN
Source of variation  Degree of freedom (DF)  Sum of Squares (SS)  Mean sum of squares (MS)  F statistic 
Treatment  t^{a}1  SST  MST = SST/ t^{a}1  MST/MSE 
Subject  n^{b}1  SSS  MSS=SSS/ n^{b}1  MSS/MSE 
Period  t1  SSP  MSP=SSP/ t1  MSP/MSE 
Error  (t1)(n2)  SSE  MSE=SSE/(t1)(n2)  
Total  tn1 
^{a}t is number of treatments; ^{b}n is number of subjects
SSTSum of squares due to treatments; SSSSum of squares due to subjects; SSPSum of squares due to period; SSE Sum of squares due to error, MST Mean sum of squares due to treatments; MSS Mean sum of squares due to subjects; MSP Mean sum of squares due to period; MES Mean sum of squares due to error
ANOVA is to be used to identify the source contributions by factors including subjects, period, formulation and potential interactions. The geometric mean ratio together with the ANOVA residual mean error term, are used to identify the statistical basis for the 90% confidence interval for the ratio of the population means (New Formulation/Original Formulation). The products were considered bioequivalent if the difference between the two compared parameters was statistically insignificant (P >0.05).
Why MANOVA over ANOVA: MANOVA is used under the same circumstances as ANOVA but when there are multiple dependent variables as well as independent variables within the model which we wish to test. MANOVA is considered as a valid alternative to the repeated measures ANOVA when sphericity is violated. In bioequivalence study we need to examine different ANOVAs for each pharmacokinetic parameter. However, since the pharmacokinetic parameters are related, the results from separate ANOVAs would not be independent. Using multiple ANOVAs would increase the risk of Type I error (rejecting the null hypothesis when it is true).
MANOVA deals with the multiple dependent variables by combining them in a linear manner to produce a combination which best separates the independent variable groups. An ANOVA is then performed on the newly developed dependent variable. In MANOVA, the independent variables relevant to each main effect are weighted to give them priority in the calculations performed. In interactions the independent variables are equally weighted to determine whether or not they have an additive effect in terms of the combined variance they account for in the dependent variables.
Like an ANOVA, MANOVA examines the degree of variance within the independent variables and determines whether it is smaller than the degree of variance between the independent variables. If the within subjects variance is smaller than the between subjects variance it means the independent variable has had a significant effect on the dependent variables.
There are two main differences between MANOVAs and ANOVAs. The first is that MANOVAs are able to take into account multiple independent and multiple dependent variables within the same model, permitting greater complexity. Secondly, rather than using the F value as the indicator of significance a number of multivariate measures (Wilks’ lambda, Pillai’s trace, Hotelling trace and Roy’s largest root) are used. The difference between the four measures is the ways in which they combine the dependent variables in order examine the amount of variance in the data.
Wilks’ lambda: Wilks’ lambda demonstrates the amount of variance accounted for in the dependent variable by the independent variable; the smaller the value, the larger the difference between the groups being analyzed. 1 minus Wilks’ lambda indicates the amount of variance in the dependent variables accounted for by the independent variables.
Pillai's trace: Pillai's trace is considered the most reliable of the multivariate measures and offers the greatest protection against Type I errors with small sample sizes. Pillai's trace is the sum of the variance which can be explained by the calculation of discriminant variables. It calculates the amount of variance in the dependent variable which is accounted for by the greatest separation of the independent variables.
HotellingLawley trace: The HotellingLawley trace is generally converted to the Hotelling’s Tsquare. Hotelling’s T is used when the independent variable forms two groups and represents the most significant linear combination of the dependent variables.
Roy’s largest root: Roy’s largest root, also known as Roy’s largest eigenvalue, is calculated in a similar fashion to Pillai's trace except it only considers the largest eigenvalue (i.e. the largest loading onto a vector). As the sample sizes increase the values produced by Pillai’s trace, HotellingLawley trace and Roy’s largest root become similar. Wilks’ lambda is the easiest to understand and therefore the most frequently used measure.
DESCRIPTION OF THE STUDY: Fenofibrate is a lipidlowering agent introduced internationally in 1975 and now used in >80 countries. It has become one of the world's most widely prescribed pharmacologic treatments for hypercholesterolemia, combined dyslipidemia, remnant hyperlipidemia, endogenous hyperlipemia (hypertriglyceridemia), and mixed hyperlipemia (Frederickson types IIa, IIb, III, IV and V dyslipidemia, respectively) ^{7, 8}.
Fenofibrate is a prodrug ^{9, 10}. After oral administration, it is rapidly converted through hydrolysis of the ester bond to its active form and major metabolite, fenofibric acid. Plasma levels of fenofibric acid peak 6 to 8 hours after oral administration, and food enhances its absorption ^{[1113]}. The extent of absorption of fenofibrate tablets is increased approximately 35% under fed as compared to fasting conditions ^{8}.
Steadystate plasma levels are reached within 5 days of dosing, and no accumulation has been observed in healthy volunteers following multiple doses ^{9}. Fenofibric acid is metabolized by the hepatic cytochrome P (CYP)450 3A4 isozyme and has a halflife (t_{1/2}) of 20 hours, which allows oncedaily administration. Fenofibrate is mainly excreted in urine as metabolites, primarily fenofibric acid and fenofibric acid glucuronide.
Since fenofibrate was first made commercially available, its main drawback has been the low bioavailability of the active metabolite, fenofibric acid, when the prodrug is taken orally on an empty stomach ^{8, 1218}. Fenofibrate is virtually insoluble in water and is highly lipophilic, hence it is poorly absorbed when taken orally, especially under fasting conditions ^{2, 9}.
In contrast, its absorption is substantially increased in the presence of food ^{7, 14, 18}. Therefore, product labeling of formulations marketed to date have mandated administering the drug with meals, even for newer fenofibrate formulations such as micronized capsule and a micro coated tablet, that were introduced to improve bioavailability ^{7,14,12} .
MATERIAL AND METHODS: The study was carried out at the B. V. Patel Pharmaceutical Education and Research Development centre, Ahmedabad. 18 subjects provided written informed consent to participate in the study prior to enrolment and were free to withdraw at any time during the study. The study was approved by the institutional ethics committee and was conducted in accordance with good clinical practice and the declaration of Helsinki.
Study Subjects: The study population consisted of 18, adult, male healthy Indian subjects with mean BMI 21.7 (range 19.14 – 24.21), a mean age of 32.2 years (range 25  44), mean weight of 59.8 kg (range 48  69) and a mean height of 165.6 cm. (range 154  177)
Design: The study was designed as Single labeled, Balanced, Randomized, Two Treatment, TwoSequence, Two Period, Single Dose, Crossover Bioequivalence study with a 14 days washout period. The volunteers were administered one of the two study drugs after standardized meal. The dose administration was performed as per the randomization schedule generated at B.V. Patel PERD Centre, Ahmedabad. Subjects received single oral doses of the test formulation (fenofibrate 145 mg) and reference formulation (fenofibrate 145 mg).
Blood sampling: A total of 16 blood samples were collected during each period. Blood samples were collected through an indwelling cannula placed in the forearm vein using disposable syringe or with disposable syringes and needles. 6 mL of blood samples (including 0.2 mL discarded heparinised blood) were withdrawn at predose and 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 10.0, 14.0, 24.0, 36.0, 48.0, 72.0 and 96.0 hrs following drug administration in each period. After centrifugation, plasma separated from blood samples and was stored at –20 ± 5°C for interim storage and then at –80 ± 4°C until analysis.
Safety and Tolerability: General clinical safety was assessed via physical examinations and vital signs conducted at screening and at the end of the study. Clinical laboratory tests and ECGs were also conducted at screening, before dosing within each treatment period, and at the end of the study. Adverse events were assessed for severity and relationship to treatment throughout the study.
Pharmacokinetic data of the study: Table 2 contains the each individual pharmacokinetic parameters of the test and reference formulation of Fenofibrate.
TABLE 2: DATA SHOWS THE PHARMACOKINETIC PARAMETERS FOR THE TEST AND REFERENCE DRUG.
Subject  A = Reference Formulation  B = Test Formulation  
C_{max}
(µg/ml) 
T_{max}
(h) 
AUC_{0t}
(µg.h/ml) 
AUC_{0∞}
(µg.h/ml) 
C_{max}
(µg/ml) 
T_{max}
(h) 
AUC_{0t}
(µg.h/ml) 
AUC_{0∞}
(µg.h/ml) 

1  9.52  6  238.98  245.35  6.16  5  153.9  158.75 
2  10.31  5  222.33  232.95  5.82  4  185.2  199.26 
3  5.17  4  104.47  107.20  5.68  4  96.8  99.21 
4  6.73  4  79.12  79.55  6.98  4  89.4  90.46 
5  5.33  3  124.01  127.02  5.47  3  112.1  117.15 
6  5.07  4  152.3  170.99  7.30  7  212.4  229.11 
7  7.30  3  136  139.49  7.06  6  130.6  133.26 
8  6.24  4  105.6  108.50  5.5  4  105.4  114.82 
9  8.37  5  192.8  208.00  5.10  5  149.4  161.50 
10  8.52  4  194.8  208.43  10.88  3  202.6  207.92 
11  10.16  5  154.6  161.77  8.32  5  152.1  159.84 
12  6.15  3  145.9  152.12  4.38  4  146.4  155.31 
13  8.18  4  105.7  106.62  5.24  5  78.7  79.39 
14  3.15  4  103.4  107.98  3.61  3  86.9  90.21 
15  3.41  4  55.3  55.74  5.25  4  83.3  83.84 
16  6.56  4  206.9  225.37  5.50  5  172.6  189.00 
17  12.13  7  275.6  294.93  10.60  4  175.5  194.86 
18  5.84  5  165.2  170.52  6.35  4  179.1  191.00 
To run MANOVA in SPSS 16.0 software dependent variables are pharmacokinetic parameters (C_{max}, T_{max}, AUC_{0t} and AUC_{0∞}) and independent (fixed) factors are drug (test/reference) and phases (phase I/phase II).
RESULTS:
ANOVA for C_{max}:
TABLE 3: DESCRIPTIVE STATISTICS OF C_{max}
Drug  Phase  Mean  Std. Deviation  N 
Reference  Phase I  6.7300  1.86439  9 
Phase II  7.5078  2.93641  9  
Total  7.1189  2.41940  18  
Test  Phase I  6.8200  2.52322  9 
Phase II  5.9800  1.03035  9  
Total  6.4000  1.91896  18  
Total  Phase I  6.7750  2.15266  18 
Phase II  6.7439  2.27488  18  
Total  6.7594  2.18280  36 
Dependent Variable: C_{max}
TABLE 4: LEVENE’S TEST OF EQUALITY OF ERROR VARIANCE^{a} FOR C_{max}
F  df1  df2  
3.781  3  32  0.020 
Dependent Variable: C_{max}; Tests the null hypothesis that the error variance of the dependent variable is equal across groups. ^{a}. Design: Intercept + Drug + Phase + Drug * Phase
TABLE 5: TESTS OF BETWEENSUBJECTS EFFECTS FOR C_{max}
Source  Type III Sum of Squares  df  Mean Square  F  
Corrected Model  10.549^{a}  3  3.516  0.720  0.547 
Intercept  1644.843  1  1644.843  336.942  0.000 
Drug  4.651  1  4.651  0.953  0.336 
Phase  0.009  1  0.009  0.002  0.967 
Drug * Phase  5.889  1  5.889  1.206  0.280 
Error  156.214  32  4.882  
Total  1811.605  36  
Corrected Total  166.762  35 
Dependent Variable:C_{max}; ^{a.} R Squared = 0.063 (Adjusted R Squared = 0.025)
ANOVA for T_{max}:
TABLE 6: DESCRIPTIVE STATISTICS OF T_{max}
Drug  Phase  Mean  Std. Deviation  N 
Reference  Phase I  4.2222  .97183  9 
Phase II  4.4444  1.13039  9  
Total  4.3333  1.02899  18  
Test  Phase I  3.8889  0.78174  9 
Phase II  4.8889  1.05409  9  
Total  4.3889  1.03690  18  
Total  Phase I  4.0556  0.87260  18 
Phase II  4.6667  1.08465  18  
Total  4.3611  1.01848  36 
Dependent Variable: T_{max}
TABLE 7: LEVENE’S TEST OF EQUALITY OF ERROR VARIANCE^{a} FOR T_{max}
F  df1  df2  
0.241  3  32  0.867 
Dependent Variable:T_{max}, Tests the null hypothesis that the error variance of the dependent variable is equal across groups. ^{a.} Design: Intercept + Drug + Phase + Drug * Phase
TABLE 8: TESTS OF BETWEENSUBJECTS EFFECTS FOR T_{max}
Source  Type III Sum of Squares  df  Mean Square  F  
Corrected Model  4.750^{a}  3  1.583  1.606  0.207 
Intercept  684.694  1  684.694  694.338  0.000 
Drug  0.028  1  0.028  0.028  0.868 
Phase  3.361  1  3.361  3.408  0.074 
Drug * Phase  1.361  1  1.361  1.380  0.249 
Error  31.556  32  0.986  
Total  721.000  36  
Corrected Total  36.306  35 
Dependent Variable: T_{max}, ^{a.} R Squared = 0.131 (Adjusted R Squared = 0.049)
ANOVA for AUC_{0t}:
TABLE 9: DESCRIPTIVE STATISTICS OF AUC_{0t}
Drug  Phase  Mean  Std. Deviation  N 
Reference  Phase I  1.4126E2  56.42855  9 
Phase II  1.6574E2  62.18373  9  
Total  1.5350E2  58.96459  18  
Test  Phase I  1.4323E2  43.27250  9 
Phase II  1.3591E2  45.50615  9  
Total  1.3957E2  43.24195  18  
Total  Phase I  1.4225E2  48.79195  18 
Phase II  1.5083E2  55.04277  18  
Total  1.4654E2  51.44736  36 
Dependent Variable: AUC_{0t}
TABLE 10: LEVENE’S TEST OF EQUALITY OF ERROR VARIANCE^{a} FOR AUC_{0t}
F  df1  df2  
0.241  3  32  0.867 
Dependent Variable: AUC_{0t}, Tests the null hypothesis that the error variance of the dependent variable is equal across groups. a. Design: Intercept + Drug + Phase + Drug * Phase
TABLE 11: TESTS OF BETWEENSUBJECTS EFFECTS FOR AUC_{0t}
Source  Type III Sum of Squares  df  Mean Square  F  
Corrected Model  4684.541^{a}  3  1561.514  0.568  0.640 
Intercept  773023.559  1  773023.559  281.245  0.000 
Drug  1745.366  1  1745.366  0.635  0.431 
Phase  662.938  1  662.938  0.241  0.627 
Drug * Phase  2276.236  1  2276.236  0.828  0.370 
Error  87954.526  32  2748.579  
Total  865662.625  36  
Corrected Total  92639.066  35 
Dependent Variable: AUC_{0t}, ^{a.} R Squared = 0.051 (Adjusted R Squared = 0.038).
ANOVA for AUC_{0∞}
TABLE 12: DESCRIPTIVE STATISTICS OF AUC_{0∞}
Drug  Phase  Mean  Std. Deviation  N 
Reference  Phase I  1.4760E2  60.32269  9 
Phase II  1.7490E2  68.35799  9  
Total  1.6125E2  64.09958  18  
Test  Phase I  1.5247E2  47.20704  9 
Phase II  1.4251E2  51.06314  9  
Total  1.4749E2  47.97921  18  
Total  Phase I  1.5004E2  52.60603  18 
Phase II  1.5871E2  60.85827  18  
Total  1.5437E2  56.23577  36 
Dependent Variable: AUC_{0∞}
TABLE 13: LEVENE’S TEST OF EQUALITY OF ERROR VARIANCE^{a} FOR AUC_{0∞}
F  df1  df2  
0.752  3  32  0.529 
Dependent Variable:AUC_{0∞}. Tests the null hypothesis that the error variance of the dependent variable is equal across groups. ^{a.} Design: Intercept + Drug + Phase + Drug * Phase
TABLE 14: TESTS OF BETWEENSUBJECTS EFFECTS FOR AUC_{0∞}
Source  Type III Sum of Squares  df  Mean Square  F  
Corrected Model  5505.433^{a}  3  1835.144  0.558  0.646 
Intercept  857913.705  1  857913.705  261.010  0.000 
Drug  1703.228  1  1703.228  0.518  0.477 
Phase  677.060  1  677.060  0.206  0.653 
Drug * Phase  3125.146  1  3125.146  0.951  0.337 
Error  105180.721  32  3286.898  
Total  968599.860  36  
Corrected Total  110686.154  35 
Dependent Variable: AUC_{0∞}. ^{a}. R Squared = 0.050 (Adjusted R Squared = 0.039)
MANOVA for all pharmacokinetic parameters:
TABLE 15: DESCRIPTIVE STATISTICS OF ALL PHARMACOKINETIC PARAMETERS
Drug  Phase  Mean  Std. Deviation  N  
AUC_{0∞}  Reference  Phase I  1.4760E2  60.32269  9 
Phase II  1.7490E2  68.35799  9  
Total  1.6125E2  64.09958  18  
Test  Phase I  1.5247E2  47.20704  9  
Phase II  1.4251E2  51.06314  9  
Total  1.4749E2  47.97921  18  
Total  Phase I  1.5004E2  52.60603  18  
Phase II  1.5871E2  60.85827  18  
Total  1.5437E2  56.23577  36  
C_{max}  Reference  Phase I  6.7299  1.86638  9 
Phase II  7.5064  2.93711  9  
Total  7.1182  2.42042  18  
Test  Phase I  6.8178  2.52154  9  
Phase II  5.9778  1.03011  9  
Total  6.3978  1.91786  18  
Total  Phase I  6.7738  2.15252  18  
Phase II  6.7421  2.27541  18  
Total  6.7580  2.18301  36  
AUC_{0t}  Reference  Phase I  1.4126E2  56.42855  9 
Phase II  1.6574E2  62.18373  9  
Total  1.5350E2  58.96459  18  
Test  Phase I  1.4323E2  43.27250  9  
Phase II  1.3591E2  45.50615  9  
Total  1.3957E2  43.24195  18  
Total  Phase I  1.4225E2  48.79195  18  
Phase II  1.5083E2  55.04277  18  
Total  1.4654E2  51.44736  36  
T_{max}  Reference  Phase I  4.2222  .97183  9 
Phase II  4.4444  1.13039  9  
Total  4.3333  1.02899  18  
Test  Phase I  3.8889  .78174  9  
Phase II  4.8889  1.05409  9  
Total  4.3889  1.03690  18  
Total  Phase I  4.0556  .87260  18  
Phase II  4.6667  1.08465  18  
Total  4.3611  1.01848  36 
TABLE 16: BOX'S TEST OF EQUALITY OF COVARIANCE MATRICES^{a }FOR ALL PHARMACOKINETIC PARAMETERS
Box's M  32.217 
F  0.822 
df1  30 
df2  2.815E3 
0.741 
Tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups. ^{a}. Design: Intercept + Drug + Phase + Drug * Phase
TABLE 17: MULTIVARIATE TESTS^{c} FOR ALL PHARMACOKINETIC PARAMETERS
Effect  Value  F  Hypothesis df  Error df  Noncent. Parameter  Observed Power^{b}  
Intercept  Pillai's Trace  0.961  1.765E2^{a}  4.000  29.000  0.000  705.805  1.000 
Wilks' Lambda  0.039  1.765E2^{a}  4.000  29.000  0.000  705.805  1.000  
Hotelling's Trace  24.338  1.765E2^{a}  4.000  29.000  0.000  705.805  1.000  
Roy's Largest Root  24.338  1.765E2^{a}  4.000  29.000  0.000  705.805  1.000  
Drug  Pillai's Trace  0.069  0.541^{a}  4.000  29.000  0.707  2.163  0.161 
Wilks' Lambda  0.931  0.541^{a}  4.000  29.000  0.707  2.163  0.161  
Hotelling's Trace  0.075  0.541^{a}  4.000  29.000  0.707  2.163  0.161  
Roy's Largest Root  0.075  0.541^{a}  4.000  29.000  0.707  2.163  0.161  
Phase  Pillai's Trace  0.141  1.192^{a}  4.000  29.000  0.335  4.769  0.325 
Wilks' Lambda  0.859  1.192^{a}  4.000  29.000  0.335  4.769  0.325  
Hotelling's Trace  0.164  1.192^{a}  4.000  29.000  0.335  4.769  0.325  
Roy's Largest Root  0.164  1.192^{a}  4.000  29.000  0.335  4.769  0.325  
Drug * Phase  Pillai's Trace  0.200  1.811^{a}  4.000  29.000  0.154  7.244  0.483 
Wilks' Lambda  0.800  1.811^{a}  4.000  29.000  0.154  7.244  0.483  
Hotelling's Trace  0.250  1.811^{a}  4.000  29.000  0.154  7.244  0.483  
Roy's Largest Root  0.250  1.811^{a}  4.000  29.000  0.154  7.244  0.483 
 Exact Statistic. ^{b.} Computed using alpha = 0.05. ^{c.} Design: Intercept + Drug + Phase + Drug*phase
TABLE 18: LEVENE'S TEST OF EQUALITY OF ERROR VARIANCES^{a} FOR ALL PHARMACOKINETIC PARAMETERS
F  df1  df2  
AUC_{0∞}  0.752  3  32  0.529 
C_{max}  3.774  3  32  0.020 
AUC_{0t}  0.700  3  32  0.559 
T_{max}  0.241  3  32  0.867 
Tests the null hypothesis that the error variance of the dependent variable is equal across groups. ^{a.} Design: Intercept + Drug + Phase + Drug * Phase
TABLE 19: TESTS OF BETWEENSUBJECTS EFFECTS FOR ALL PHARMACOKINETIC PARAMETERS
Source  Dependent Variable  Type III Sum of Squares  df  Mean Square  F  Noncent. Parameter  Observed Power^{b}  
Corrected Model  AUC_{0∞}  5505.433^{a}  3  1835.144  0.558  0.646  1.675  0.152 
C_{max}  10.560^{c}  3  3.520  0.721  0.547  2.163  0.186  
AUC_{0t}  4684.541^{d}  3  1561.514  0.568  0.640  1.704  0.154  
T_{max}  4.750^{e}  3  1.583  1.606  0.207  4.817  0.381  
Intercept  AUC_{0∞}  857913.705  1  857913.705  261.010  0.000  261.010  1.000 
C_{max}  1644.127  1  1644.127  336.751  0.000  336.751  1.000  
AUC_{0t}  773023.559  1  773023.559  281.245  0.000  281.245  1.000  
T_{max}  684.694  1  684.694  694.338  0.000  694.338  1.000  
Drug  AUC_{0∞}  1703.228  1  1703.228  0.518  0.477  0.518  0.107 
C_{max}  4.671  1  4.671  0.957  0.335  0.957  0.158  
AUC_{0t}  1745.366  1  1745.366  0.635  0.431  0.635  0.121  
T_{max}  .028  1  0.028  0.028  0.868  0.028  0.053  
Phase  AUC_{0∞}  677.060  1  677.060  0.206  0.653  0.206  0.072 
C_{max}  .009  1  0.009  0.002  0.966  0.002  0.050  
AUC_{0t}  662.938  1  662.938  0.241  0.627  0.241  0.076  
T_{max}  3.361  1  3.361  3.408  0.074  3.408  0.433  
Drug * Phase  AUC_{0∞}  3125.146  1  3125.146  0.951  0.337  0.951  0.157 
C_{max}  5.880  1  5.880  1.204  0.281  1.204  0.187  
AUC_{0t}  2276.236  1  2276.236  0.828  0.370  0.828  0.143  
T_{max}  1.361  1  1.361  1.380  0.249  1.380  0.207  
Error  AUC_{0∞}  105180.721  32  3286.898  
C_{max}  156.234  32  4.882  
AUC_{0t}  87954.526  32  2748.579  
T_{max}  31.556  32  0.986  
Total  AUC_{0∞}  968599.860  36  
C_{max}  1810.920  36  
AUC_{0t}  865662.625  36  
T_{max}  721.000  36  
d  AUC_{0∞}  110686.154  35  
C_{max}  166.794  35  
AUC_{0t}  92639.066  35  
T_{max}  36.306  35 
 R squared = 0.050 (Adjusted R squared = 0.039). ^{b.} Computed using alpha = 0.05. ^{c.} R squared = 0.063 (Adjusted R squared = 0.025). ^{d.} R squared = 0.051 (Adjusted R squared = 0.038). ^{e.} R squared = 0.131 (Adjusted R squared = 0.049)
DISCUSSION: Table 3, 6, 9, 12, 15 provides the mean and standard deviation for the groups that have been split by both independent variables. In addition, the tables also provide “total” rows, which allow means and standard deviations for groups only split by one independent variable for all dependent variables.
ANOVAMANOVA comparison: Table 4, 7, 10, 13 and 18 shows Levene’s test of equality of error variances of all dependent variables. Levene’s test and Box’s M test are almost same but the only difference is this test is concern about variance only. From table 4 we can see that we have homogeneity of variances of the dependent variables across groups. Here Sig. = 0.020 < 0.05 (level of alpha), so from this we can say that the variance across groups was significantly different for dependent variables. From table 7, 10 and 13 we have sig. = 0.867 > 0.05, sig. = 0. 559 > 0.05 and sig. = 0.529 > 0.05, so we can say that the variance across groups was not significantly different for dependent variable T_{max},AUC_{ot}, AUC_{o∞}. Same values we have in MANOVA analysis table 18 for all the dependent variables.
In bioequivalence study, instead of doing different ANOVAs for pharmacokinetic parameters we can do MANOVA and have the same results like ANOVA. Further MANOVA has four tests, from that we can interpret more our data instead of ANOVA. Table 5, 8, 11, 14 shows the test of betweensubject effects (ANOVA) and Table no. 19 shows the ANOVA results from MANOVA analysis for all dependent variables. Table 5, 8, 11, 14 indicate that whether significant mean differences between groups for two independent variables (drug and phase) and for their interaction (drug*phase) for all dependent variables. From Table 5, 8, 11 and 14 we can say that drug*phase interaction have a statistically significant interaction at the p=0.280 level, p = 0.249, p = 0.370 and p = 0.337 respectively.
We can say from tables, there was no significant difference in dependent variables between two drugs (p = 0.336>0.05), (p = 0.868>0.05), (p = 0.431>0.05) and (p = 0.477>0.05) respectively and similarly for phases (p = 0.967>0.05), (p = 0.074>0.05), (p = 0.627>0.05) and (p = 0.653>0.05). From Table 19 tests betweensubjects effect for all pharmacokinetic parameters we can see four dependent variables, F column shows the value of F ratio and Sig. column shows the significance of that F ratio. So comparing this table with the Table no. 5, 8, 11 and 14 we have the same results.
Multivariate Tests Analysis: Table 16 shows Box’s Test of equality of covariance matrices. This test in effect asks whether the correlations between the dependent variables and the standard deviations are similar over groups.
In this table the Box’s test is not significant, so the variancecovariance matrices can be pooled without any concern. Here sig. = 0.741>0.05. Therefore, the variancecovariance matrices are equal.
Table 17 shows Multivariate test of the analysis. All 4 tests explore whether the means for phases and drugs are the same or not. Among 4 tests the most commonly used and accepted statistic is Wilk’s Lambda. It is a statistics to test whether there are differences between the means of identified groups of subjects on a combination of dependent variables. Ttest, Hotelling’s T and Ftest are special cases of Wilk’s Lambda.
It is a measure of the percent of variance in the dependent variables that is not explained by differences in the level of the independent variables (drug and phase).
Here we have λ=0.931, F (4, 29) = 0.541, P (0.707)>0.001 for drug, for phase λ=0.859, F (4, 29) = 1.192, P (0.335)>0.001, and for interaction term (drug*phase) we have λ=0.800, F (4, 29) = 1.811, P (0.154)>0.001. Therefore, from this result we can say that 6.9%, 14.1% and 20% of the variance of the dependent variable is accounted for by the differences between drugs, phase and interaction respectively.
The value of Pillai’s Trace is a positive valued statistic and it shows the proportion of variance in the dependent variables which is accounted for by variation in the independent variables. Here we have 0.069, 0.141 and 0.200 which is very small value that lead to statistical insignificance.
Hotelling’s Trace is the sum of the eigen values of the test matrix and it is a positive valued statistic for which increasing values indicate effects that contribute more to the model. Roy’s largest root is similar to the Pillai’s trace but is based only on the first root. It is less robust than the other tests in the face of violations of the assumptions of multivariate normality.
Same like other tests larger the root, the more that effect contributes to the model. Here we have Hotelling’s trace and Roy’s largest root’s values are 0.075, 0.164 and 0.250 for drug, phase and interaction respectively. This shows smaller values that lead to statistical insignificance.
CONCLUSION: The concept of BE has been accepted worldwide by the pharmaceutical industry and national regulatory authorities for over 20 years and is applied to new as well as generic products. As a result, thousands of highquality generic drugs at reduced costs have become available in every corner of the globe.
The assessment of BE is not a simple issue, however, and much of the research has been done in recent years to develop new and more effective approaches to the assessment of BE. Statistical analysis is a part of BE and we need to abridge it in such a way that it involve less time and more construal from the data.
The essential feature of doing MANVOA is we have complete ANOVA results and adding the multivariate results. So from that we can check the significance of the dependent variables.
Additionally we get multivariate analysis. The value of Wilk’s Lambda shows the proportion of the total variance of the dependent variable which is not accounted for by the independent variables. Therefore, smaller the value of Lambda corresponds to larger differences between groups (or strong associations between the dependent variables and numeric independent variables). Here we have larger values of Wilks lambda that shows minimum differences between groups or we can say weak association between the dependent variables and independent variables.
If Pillai’s Trace has large value then the more the given effect contributes to the model. In other words same like Hotelling’s trace and Roy’s largest root, increasing values of the statistic indicate effects that contribute more to the model. So here we have small value of all the three tests shows statistical insignificance to the model.
So now we can say that we can use MANOVA instead of doing separate ANOVA. And we can control the increase the risk of Type I error.
ACKNOWLEDGEMENT: The authors thank Troikaa Pharmaceuticals Ltd, Ahmedabad, India, for providing the drug samples, and we are also grateful to clinical staff of B. V. Patel PERD Centre for their assistance in the pharmacokinetic study.
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How to cite this article:
Patel S, Padh H and Bhavsar C: “MANOVA over ANOVA” A Better Objective in Bioequivalence Study. Int J Pharm Sci Res 2013; 4(5); 18741884.
Article Information
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IJPSR
S. Patel*, H. Padh and C. Bhavsar
Research Scholar, Department of Statistics, University School of Sciences, Gujarat University 1, Navrangpura, Ahmedabad380009, Gujarat, India
sweta1611us@gmail.com
08 January, 2013
27 February, 2013
25 April, 2013
http://dx.doi.org/10.13040/IJPSR.09758232.4(5).187481
01 May, 2013