6-Statistical Methods in Bioefficicacy Trials

8/3/2019 6-Statistical Methods in Bioefficicacy Trials

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SOME STATISTICAL TECHNIQUES FOR BIO-EFFICACY TRIALS

Rajender Parsad

I.A.S.R.I., Library Avenue, New Delhi - 110 012

Biological assays and probit analysis are quite useful in designing of bio-efficacy trialsand analysis of experimental data. A biological assay (bioassay) is a set of techniques

relevant to the comparisons between the biological strength of alternative but similar

biological stimuli (for example, a vitamin, a drug, a mental test, a physical force, aninsecticide, plant extract, etc.) based on the responses produced by them on the subjects,

such as subhuman primates' (or human) living tissues, plants or isolated organs, insects,

etc. Normally two preparations of the stimulus, one of known strength (standardpreparation) and another of unknown strength (test preparation), both with quantitativedoses are applied to a set of living organisms. The general objective of bioassays is to

draw statistically valid conclusions on the relative potency of the test preparation with

respect to the standard one. Usually when a drug or a stimulus is applied to a subject itinduces a change in some measurable characteristic that is designated as the response

variable. In this setup, the dose may have several chemically or therapeutically different

ingredients, while the response may also be multivariable. Thus the stimulus-response ordose-response relationships for the two preparations, both subject to inherent stochastic

variability, are to be compared in a sound statistical manner so as to cast light on their

relative performance with respect to the set objectives.

Let ds and dtdenote the doses of the standard and the test preparations respectively such

that each of them produces a pre-assigned response in some living organism. Then the

ratio ts dd= is called the relative potency of the test preparation. If is greater than

unity, it shows that a smaller dose of the test preparation produces as much response as arelatively larger dose of the standard preparation and hence the potency of the test

preparation is greater than that of the standard preparation. Similarly when is less thanunity the potency of the test preparation is smaller than that of the standard preparation.

Naturally, such statistical procedures may depend on the nature of the stimulus andresponse, as well as on other extraneous experimental (biological or therapeutic)

considerations. As may be the case with some competing chemicals for removing the

effect of common insects, the two (i.e. test and standard) preparations may not have thesame chemical or pharmacological constitution, and hence, statistical modeling may need

a somewhat different approach than in common laboratory experimentation.

Nevertheless, in many situations the test preparation may behave (in terms of theresponse/tolerance distribution) as if it is a dilution or concentration of the standard one.For this reason, often such bioassays are designated to compare the relative performance

of two drugs under the dilution-concentration postulation (i.e., assays with two

preparations containing the same effective ingredients which is responsible for theresponse), and are thereby termed dilution assays or analytical dilution assays. Dilution

assays are classified into two broad categories: direct dilution and indirect dilution. In a

direct assay, for each preparation the exact amount of dose needed to produce a specified


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Some Statistical Techniques for Bio-Efficacy Trials

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response is directly measured, so that the response is certain while the dose is a non-negative random variable that defines the tolerance distribution. Statistical modeling of

these tolerance distributions enables us to define the relative potency in a statistically

interpretable and analyzable manner, often in terms of the parameters associated with thetolerance distributions. However, a direct assay is practicable only when both the

preparations are capable of administration in such a way that the minimal amountsneeded to produce the specified response can directly be measured. In most assays,however, the response is not directly measurable and indirect methods are used to

estimate the dose corresponding to a given response via a dose response relationship,

such assays are known as indirect assays. In an indirect assay the dose is generally

administered at some prefixed (usually non-stochastic) levels, and at each level theresponse is observed for subjects included in the study. Thus, the dose is generally non-

stochastic and the stochastic response at each level provides information about the

tolerance distribution for the particular preparation. If the response is a quantitativevariable (magnitude of some property like survival time, weight, etc.), then we have an

indirect quantitative assay, while if the response is quantal in nature (i.e. all or nothing),

then we have a quantal assay. Both of these assays are commonly adopted in statisticalpractice. Within this framework, the nature of the dose-response regression may call for

suitable transformations on the dose variable (called the dosage or dosemetameter)

and/or the response variable, called the response-metameter. The basic objective of such

transformations is to achieve a linear dose-response regression that may inducesimplifications in statistical modeling and analysis schemes. Ifz represents the dose in

the original scale, then the two transformations that have been found useful in bioassay

work are (i) )z(logx e= and (ii)zx = , where 0> is a known constant. The first of

these gives rise toparallel line assays and assays based on the second transformation are

called as slope ratio assays. These assays normally fall under the category of

quantitative indirect assays. In these assays, the transformation of response variable is

generally not needed. In quantal assays, however, the response variable is generallysubjected to theprobit(or normit) and logittransformations, based on normal and logistic

distributions, respectively.

In a parallel line assay, the two dose-response regression lines (one for the standard

preparation and another for the test preparation) are taken as parallel and further the

errors in the two regression equations are assumed to have the same distribution (oftentaken as normal). Therefore, after fitting of these two regression lines, it is important to

test for the parallelism for the regression lines before making any conclusions. A

parallel line assay is called as symmetric if the standard and test preparations involve thesame number of doses, otherwise it is called an asymmetric assay. The analysis of the

parallel line assay for conducting the validity tests and for estimating the relative potencybecomes very much simplified when the doses of each of the preparations are taken in

geometric progression.

In slope ratio assays it is assumed that the two regression lines intersect at the same point

on the response axis, i.e., they are assumed to have the same intercept. As the dose takesvalue zero on the response axis, it is necessary to include a blank dose in the assay for the

validity test. Thus if there are k doses for each of the two preparations in a slope ratio


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assay, the total number of doses is 2k+1. If against each of these doses an equal numberof subjects, say, n is allotted, the assay is 2k+1 symmetrical slope ratio assay, otherwise

asymmetrical slope ratio assays. In slope ratio assays, the blank, the intersection

contrasts and the two regression contrasts are of major importance.

In quantal assays, occurrence or non-occurrence will depend upon the intensity of thestimulus. For any one subject, under controlled conditions there will be a certain level ofintensity below which the response does not occur and above which the response occurs.

Such a value has often been called a thresholdor limen, but the term tolerance is now

widely accepted. This tolerance value will vary from one member to another of the

population used, frequently between quite wide limits. When the characteristic responseis quantitative, the stimulus intensity needed to produce a response of any given

magnitude will show similar variation between individuals. In either case, the value for

an individual also is likely to vary from one occasion to another as a result ofuncontrolled internal or external condition.

In these assays, the earlier attempts were made to characterize the effectiveness of astimulus in relation to a quantal response referred to the minimal effective dose or for a

more restricted class of stimuli, the minimal lethal dose terms which failed to take

account of the variation in tolerance within a population. The logical weakness of such

concepts is the assumption that there is a dose for any given chemical which is only justsufficient to kill all or most of the insects of a given species, and, doses a bit lesser would

not kill any insect of that species. Any worker, however, accustomed to the estimation of

toxicity knows that these assumptions do not represent the truth.

It might be thought that the minimal lethal dose of a poison could instead be defined asthe dose just sufficient to kill a member of the species with the least possible tolerance,

and also a maximal non-lethal dose as the dose which will just fail to kill the most

resistant member. Undoubtedly some doses are so low that no test subject will succumbto them and others so high as to prove fatal at all, but considerable difficulties attend

determination of the end-points of these ranges. Even when the tolerance of an individual

can be measured directly, to say from measurements on a sample of ten or a hundred thatthe lowest tolerance found indicated the minimal lethal dose would be unwise: a larger

sample might contain a more extreme member. When only quantal responses for selected

doses can be recorded the difficulty is increased, and the occurrence of exceptional

individuals in the batches at different dose levels may seriously bias the final estimates.The problem is in fact that of determining the dose at which the dose response curve for

the whole population needs the 0% or 100% levels of kill and even a very large

experiment could scarcely estimate these points with any accuracy.

An escape from the dilemma can be made by giving attention to a different and more

satisfactorily defined characteristics, the median lethal dose, or, as a more general term toinclude response other then death, the median effective dose. This is the dose that will

produce a response in half the population. The median effective dose is commonly

referred to as the ED 50, the more restricted concept of median lethal dose as the LD 50.Analogous symbols were used for doses effective for other proportions of the population,


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ED90 being the dose that causes 90% to respond. With a fixed total number of subjecteffective doses in the neighborhood of ED50 can usually be estimated more precisely

then those for more extreme percentage levels and this is, therefore, particularly favoured

in expressing the effectiveness of the stimulus. The ED50 alternatively be regarded asthe median of the tolerance distribution that is to say the level of tolerance such that

exactly half the subjects lie on either side of it.

The ED 50 or LD 50 can easily be calculated using the Probit Analysis. For doing this,

we conduct an experiment on different doses of an insecticide applied under standardized

conditions to samples of an insect species and record the number of insects killed and the

number of insects exposed. Now the ratio of the number insects died (the subjectsresponded) to that of the number of insects exposed (subjects exposed) gives the

probability or proportion (P) of the insects killed at a particular dose. Now this

probability data is subjected to probit or logit transformation.

Probit transformation is nothing but the 5 more than the normal equivalent deviate. In

this transformation, we replace each of the observed proportions with the value ofstandard normal curve below which the observed proportion of the area is found. To

avoid negative numbers, the constant 5 is usually added. For example, if half (0.5) of the

subjects respond at a particular dose, the corresponding probit value is 0, since half of the

area in a standard normal falls below a Z score of 0. When the constant 5 is added, thetransformed value for the proportion is 5. If the observed proportion is 0.95, the

corresponding probit value is 1.64. Addition of the constant value of 5 makes this 6.64.

Likewise, if 10% of the subjects respond, then the normal equivalent deviate is -1.29 andhence the probit value is 3.7.

In the logit transformation, the observed proportion P is changed to

( )( ) 52

1ln+

PP

The quantity ( )( )PP 1ln is called a logit. Division by 2 and addition of the constant 5 isdone to keep the values positive and to keep the two types of transformations on a similar

scale. If the observed proportion is 0.5, the logit-transformed value is 0+5, the same as

the probit-transformed value. Similarly, if the observed proportion is 0.95, the logit-transformed value is 6.47 (1.47+5). This differs somewhat from the corresponding probit

value of 6.64. (In most situations, analyses based on logits and probits give very similar

results.)

The above is discussion about the transformation of the observed proportions. The dose

are transformed to the logarithmic scale. When the experimental data on the relation

between dose and proportion of the subjects responded have been obtained, either agraphical or a statistical approach in terms of fitting of response metameter-dose

metameter linear regression relationship can be used to estimate the parameters.


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The graphical approach is rapid and sufficiently good for many purposes. In thisapproach, the probits are plotted against the dose-metameter, and a straight line is drawn

by eye to fit the points as satisfactorily as possible. The line must be so drawn that the

differences between the observed probit values and the probit values obtained by the lineat each value of dose metameter are as small as possible. The value of log ED 50 is

estimated from this line asx at which probit value is 5. In the second approach, we fit asimple linear regression equation

Probit ( iii ebxaP ++=) ,

where iP is the observed proportion corresponding at close ix (usually the log of the dose

is used instead of the actual dose), a and b are respectively the intercept and slope of the

regression equation and ie is the random error. From the fitted regression equation, the

log ED 50 (x) is obtained as that value for which probit value is 5.

The goodness of fit of the model, the Pearsons chi-square goodness of fit test. For this

we obtain the expected frequencies are obtained. The expected frequencies are the

number of subjects responded and is obtained by obtaining the estimated Probit ( iP ) from

the fitted model. Subtract 5 from he model. Now obtain the area under the normal curve

below this point. This gives the percentage of the subjects responded. Now using thenumber of subjects tried, one can obtain the expected number of subjects responded

( ii Pn ) for the dose i. If ir is the corresponding observed number of the subjects affected,

then we obtain, residuals as iii Pnr . Now the Pearson goodness of fit chi-square is

obtained as)1(

)( 2

iii

iii

PPn

Pnr

. The degrees of freedom are equal to the number of doses

minus he number of estimated parameters. To be clearer, consider the following examplefrom Finney (1971).

Example 1: Finney (1971) gave a data representing the effect of a series of doses of

carotene (an insecticide) when sprayed on Macrosiphoniella sanborni (some obscure

insects). The Table below contains the concentration, the number of insects tested ateach dose, the proportion dying and the probit transformation (probit+5) of each of the

observed proportions.

Concentration

(mg/1)

No. of

insects (n)

No. of

affected (r)

%kill (P) Log

concentration

(x)

Empirical

probit

10.2 50 44 88 1.01 6.18

7.7 49 42 86 0.89 6.08

5.1 46 24 52 0.71 5.053.8 48 16 33 0.58 4.56

2.6 50 6 12 0.41 3.82

0 49 0 0 - -


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The above discussion relates to only one stimulus. If there is more than one stimulusvariables terms are added to the model for each of the stimuli. We get the regression

coefficients and standard errors, intercept and standard error, Pearson goodness-of-fit chi-

square, observed and expected frequencies, and confidence intervals for effective levelsof independent variable(s). If the Pearson goodness-of-fit chi-square is non-significant,

we calculate the value of the dose-metameter for which the value of the responsemetameter (probit) is 5. The antilogarthmic of this dose metameter is the ED 50 or LD50.

The ED 50 alone does not fully describe the effectiveness of the stimulus. Two

insecticides/fungicides may require the same rate of application in order to be lethal tohalf of the population, but, if the distribution of tolerances has a lesser 'spread' for one

than for the other, any increase or decrease from this rate will produce a greater change in

mortality for the first than for the second. Therefore, it is necessary to give the standarderrors and fiducial limits associated with ED 50 or LD 50. Besides knowing the ED 50 of

a particular chemical preparation, the experimenter may be interested in comparing the

relative potencies of the several chemical preparations. One is required to fit the probitregression lines for each of the chemical preparations separately. These regression lines

are required to be tested for parallelism. If the probit regression lines are parallel for the

different chemical preparations, then the relative potency is constant at all levels of theresponse.

The choice of an efficient experimental design is based on the nature of the variability in

the experimental material, environmental conditions and objectives for conducting abioassay. The design may be a randomized complete block design, an incomplete block

design, design for factorial experiments etc.

Computation of corrected efficacy %

The discussion earlier assumed that the responses of the test subjects is due to the appliedstimuli alone. In some experiments, however, the responses can occur at zero dose; either

control batches of the subjects have received zero dose or a sequence of low doses

indicating a minimal response rate greater than zero. In a pesticide trials some insectsmay die from natural causes. In such situations, it is required to work with crrected

mortality or corrected proportions of the responses. Corrected efficacy % in pesticide

trials can be computed by using the Abbott, Henderson and Tilton, Schneider-Orelli orSun-Shepard formulas. The selection of appropriate formula is depending on two factors

viz.

1. Trial condition (infestation or population stability and homogeneity).2. The data on your hand (live individuals or mortality %).

The following table is of help in choosing the right formula

Available data Uniform population Non-uniform population

Infestation or live

individualsAbbott Henderson-Tilton

Mortality or dead

individualsSchneider-Orelli Sun-Shepard


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These formulae are given in the sequel.

Abbott's formula

100*1

=

treatmentafterCoinn

treatmentafterTinn%Corrected

where : n = Insect population, T = treated, Co =control

Henderson Tilton's formula

100*treatmentbeforeTinn*treatmentafterCoinn

treatmentafterTinn*treatmentbeforeCoinn1%Corrected

=

where : n = Insect population, T= treated, Co =control

The above two formulae can be combined into one and written as

Corrected % = 100

x

yx

where x = % survivorship in the control group (concentration of pesticide=0)

y = % survivorship in the experimental group.

Schneider-Orelli's formula

100*controlin%Mortality-100

controlin%Mortality-in treatedMortality%%Corrected

=

Sun-Shepard's formula

100*controlin%Change100

controlin%changein treatedMortality%%Corrected

+

+=

If the response is other than the death, then we may replace mortality by responded andsurvivorship by non-responded. Once the corrected responded % is obtained, then same

procedure as above may be adoted.

{This note is prepared from the book Probit Analysis by D. J. Finney (1971)}.

Steps for carrying out the Probit Analysis using MINITAB

For the data given in example 1, first enter the data in the Worksheet of MINITAB inthree coumns C1: dose; C2: total Insects; C3: Insects killed or affected. Now create a

column C4 for logdose by using LOGT(C1) using menu Calc.

Now Choose Stat > Reliability/Survival > Probit Analysis.

From the dialog box; Choose the data format "Success/trial" or "Response/frequency". In

the present case, the data is in success trial format, therefore, enter C3, the column


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containing the number of successes in Number of Successes box and C2, the total numberof trials in Number of Trials subbox. In the subbox for stress/stimulus enter C4, the

column containing the logdose. Since, there is only one stimulus, therefore, the subbox

pertaining to Factor (optional) may be left blank. Choose the distribution as normal.

The other options available on the dialog box are: Estimate, Graphs, Options, Results and

Storage.

Using the option Estimate, One can

- estimate percentiles for the percents you specify. These percentiles are added to thedefault table of percentiles.

- estimate survival probabilities for the stress values you specify.

One can also change the method of estimation for the confidence intervals and the levelof confidence. The default option is two sided 95% fiducial intervals.

Other options may also be used, as and when required. For this example, we chose the

additional percentiles as 65 and survival probabilities for stress level 0.9 (logdose).

Probit Analysis: affect, total versus logdoseDistribution: Normal

Response Information

Variable Value Count

affect Success 132

Failure 111

total Total 243

Estimation Method: Maximum Likelihood

Regression Table

Standard

Variable Coef Error Z P

Constant -2.88746 0.350134 -8.25 0.000

logdose 4.21320 0.478303 8.81 0.000

Log-Likelihood = -120.052

Goodness-of-Fit Tests

Method Chi-Square DF P

Pearson 1.72888 3 0.631

Deviance 1.73897 3 0.628

Tolerance Distribution: Parameter Estimates

Standard 95.0% Normal CI

Parameter Estimate Error Lower Upper

Mean 0.685338 0.0220962 0.642030 0.728646

StDev 0.237349 0.0269451 0.190001 0.296497


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Table of Percentiles


Percent Percentile Error Lower Upper

1 0.133180 0.0686394 -0.0013503 0.267711

2 0.197882 0.0617254 0.0769020 0.318861

3 0.238933 0.0573944 0.126442 0.351423

4 0.269813 0.0541723 0.163638 0.375989

5 0.294933 0.0515787 0.193840 0.396025

6 0.316313 0.0493935 0.219504 0.413123

7 0.335060 0.0474969 0.241967 0.428152

8 0.351845 0.0458160 0.262047 0.441643

9 0.367110 0.0443030 0.280278 0.453943

10 0.381162 0.0429251 0.297031 0.465294

20 0.485580 0.0332991 0.420314 0.550845

30 0.560872 0.0274617 0.507048 0.614696

40 0.625206 0.0238086 0.578542 0.671870

50 0.685338 0.0220962 0.642030 0.728646

60 0.745470 0.0224241 0.701519 0.789420

65 0.776793 0.0233958 0.730939 0.822648

70 0.809804 0.0249330 0.760936 0.858672

80 0.885096 0.0299366 0.826422 0.943771

90 0.989513 0.0389715 0.913131 1.06590

91 1.00357 0.0402991 0.924581 1.08255

92 1.01883 0.0417626 0.936978 1.10068

93 1.03562 0.0433947 0.950564 1.12067

94 1.05436 0.0452427 0.965688 1.14304

95 1.07574 0.0473792 0.982882 1.16860

96 1.10086 0.0499232 1.00301 1.1987197 1.13174 0.0530936 1.02768 1.23580

98 1.17279 0.0573685 1.06035 1.28523

99 1.23750 0.0642153 1.11164 1.36336

Table of Survival Probabilities

95.0% Normal CI

Stress Probability Lower Upper

0.9 0.182888 0.122757 0.258650

Interpretation: The goodness-of-fit tests (p-values = 0.631, 0.628) suggest that the

distribution and the model fits the data adequately. In this case, the fitting is done onnormal equivalent deviate only without adding 5. Therefore, log LD50 or lof ED50

corresponds to the value of Probit=0.Log LD50 is obtained as 0.685338. Therefore, the

stress level at which the 50% of the insects will be killed is (100.685338=4.845 mg/l).

Similarly the stress level at which 65% of the insects will be killed is (100.776793

= 5.981mg/l). At logdose = 0.9, what percentage of insects will be killed? Results indicate that

18.29% of the insects will be killed.


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Example 2: {Finney (1971): Ex.13} reported an experiment conducted to compare therelative potencies of the three analgesic amidone, phenadoxone and pethidine with

morphine. The technique was to record how many standard electric shocks could be

applied to the tail of a mouse before the mouse squeaked; if the number of shocksincreased by 4 or more after application of the drug, then we say that the mouse has

responded. The data generated alongwith the log dose concentration is given thefollowing table.

Factor Log dose Total Mouse Observed Mouse Responded

(f) (x) (n) (r)

1 0.18 103 191 0.48 120 53

1 0.78 123 83

2 0.18 60 142 0.48 110 54

2 0.78 100 81

3 -0.12 90 313 0.18 80 543 0.48 90 80

4 0.70 60 13

4 0.88 85 274 1.00 60 32

4 1.18 90 55

4 1.30 60 44

1: morphine; 2:amidone; 3: phenadoxone; 4: pethidine.

For the analysis of data follow the same steps as in Example 1 with the addition that in

the factor subbox define factor as f. The results obtained are given in the sequel.


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Probit Analysis: r, n versus x, f

Distribution: Normal

Response Information

Variable Value Count

r Success 640

Failure 591

n Total 1231

Factor Information

Factor Levels Values

f 4 1, 2, 3, 4

Estimation Method: Maximum Likelihood

Regression Table

Standard

Variable Coef Error Z P

Constant -1.38969 0.114661 -12.12 0.000

x 2.47552 0.173176 14.29 0.000

f

2 0.237877 0.108353 2.20 0.028

3 1.35900 0.129801 10.47 0.000

4 -1.18220 0.132956 -8.89 0.000

Test for equal slopes: Chi-Square = 1.54180 DF = 3 P-

Value = 0.673

Log-Likelihood = -729.327

Multiple degree of freedom test

Term Chi-Square DF P

f 185.015 3 0.000

Goodness-of-Fit Tests

Method Chi-Square DF P

Pearson 4.03105 9 0.909

Deviance 4.03534 9 0.909


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f = 1: Tolerance Distribution

Parameter Estimates



Mean 0.561374 0.0291474 0.504247 0.618502

StDev 0.403956 0.0282589 0.352198 0.463319


Standard 95.0% Fiducial CI


1 -0.378367 0.0689837 -0.533080 -0.257945

2 -0.268249 0.0620774 -0.407031 -0.159538

3 -0.198383 0.0578001 -0.327263 -0.0968963

4 -0.145825 0.0546499 -0.267390 -0.0496399

5 -0.103073 0.0521384 -0.218789 -0.0111000

6 -0.0666851 0.0500422 -0.177503 0.0217857

7 -0.0347796 0.0482399 -0.141375 0.0506903

8 -0.0062121 0.0466576 -0.109088 0.0766335

9 0.0197689 0.0452472 -0.0797814 0.100284

10 0.0436845 0.0439754 -0.0528570 0.122108

20 0.221397 0.0355803 0.145115 0.286370

30 0.349540 0.0312437 0.284438 0.408244

40 0.459033 0.0292557 0.400035 0.515829

50 0.561374 0.0291474 0.504555 0.619913

60 0.663715 0.0307532 0.605621 0.727451

70 0.773209 0.0340907 0.710494 0.845760

80 0.901352 0.0395535 0.830116 0.987334

90 1.07906 0.0488712 0.992544 1.1871491 1.10298 0.0502282 1.01420 1.21423

92 1.12896 0.0517234 1.03768 1.24371

93 1.15753 0.0533905 1.06346 1.27616

94 1.18943 0.0552784 1.09219 1.31246

95 1.22582 0.0574616 1.12491 1.35392

96 1.26857 0.0600631 1.16327 1.40270

97 1.32113 0.0633085 1.21034 1.46276

98 1.39100 0.0676909 1.27277 1.54273

99 1.50112 0.0747255 1.37093 1.66903


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Parameter Estimates



Mean 0.465283 0.0329958 0.400612 0.529953

StDev 0.403956 0.0282589 0.352198 0.463319




1 -0.474459 0.0766704 -0.646325 -0.340552

2 -0.364341 0.0697856 -0.520318 -0.242102

3 -0.294474 0.0655120 -0.440558 -0.179453

4 -0.241917 0.0623566 -0.380675 -0.132207

5 -0.199165 0.0598338 -0.332051 -0.0936894

6 -0.162777 0.0577218 -0.290734 -0.0608351

7 -0.130871 0.0558997 -0.254566 -0.0319695

8 -0.102304 0.0542942 -0.222234 -0.0060724

9 -0.0763229 0.0528573 -0.192875 0.0175260

10 -0.0524073 0.0515559 -0.165892 0.0392905

20 0.125305 0.0427040 0.0329891 0.202644

30 0.253448 0.0376004 0.173847 0.322983

40 0.362942 0.0345099 0.291660 0.428352

50 0.465283 0.0329958 0.399014 0.529602

60 0.567624 0.0330004 0.503306 0.633914

70 0.677117 0.0346823 0.611508 0.748895

80 0.805260 0.0385371 0.734347 0.887252

90 0.982973 0.0462875 0.899894 1.0839491 1.00689 0.0474826 0.921870 1.11071

92 1.03287 0.0488128 0.945680 1.13986

93 1.06144 0.0503107 0.971790 1.17198

94 1.09334 0.0520233 1.00087 1.20793

95 1.12973 0.0540229 1.03395 1.24902

96 1.17248 0.0564284 1.07270 1.29741

97 1.22504 0.0594584 1.12019 1.35705

98 1.29491 0.0635920 1.18312 1.43653

99 1.40502 0.0703037 1.28191 1.56220


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Parameter Estimates



Mean 0.0123990 0.0356207 -0.0574164 0.0822144

StDev 0.403956 0.0282589 0.352198 0.463319




1 -0.927342 0.0819421 -1.11109 -0.784282

2 -0.817224 0.0750288 -0.985032 -0.685889

3 -0.747358 0.0707272 -0.905217 -0.623295

4 -0.694800 0.0675438 -0.845278 -0.576104

5 -0.652049 0.0649928 -0.796599 -0.537642

6 -0.615660 0.0628521 -0.755225 -0.504845

7 -0.583755 0.0610006 -0.718999 -0.476037

8 -0.555188 0.0593651 -0.686607 -0.450199

9 -0.529206 0.0578974 -0.657187 -0.426662

10 -0.505291 0.0565644 -0.630141 -0.404960

20 -0.327579 0.0473402 -0.430521 -0.242346

30 -0.199436 0.0417348 -0.288661 -0.123009

40 -0.0899420 0.0379801 -0.169516 -0.0189715

50 0.0123990 0.0356207 -0.0604621 0.0805777

60 0.114740 0.0346077 0.0458819 0.182837

70 0.224234 0.0351438 0.156392 0.295510

80 0.352377 0.0377872 0.281665 0.431433

90 0.530089 0.0442858 0.449713 0.62562091 0.554004 0.0453498 0.471950 0.652131

92 0.579986 0.0465463 0.496026 0.681012

93 0.608553 0.0479071 0.522408 0.712859

94 0.640458 0.0494781 0.551772 0.748529

95 0.676847 0.0513295 0.585143 0.789329

96 0.719598 0.0535776 0.624204 0.837409

97 0.772156 0.0564359 0.672038 0.896705

98 0.842022 0.0603728 0.735353 0.975800

99 0.952140 0.0668332 0.834640 1.10097


16/16


VI-76


Parameter Estimates



Mean 1.03893 0.0281126 0.983832 1.09403

StDev 0.403956 0.0282589 0.352198 0.463319




1 0.0991908 0.0704720 -0.0590689 0.222048

2 0.209309 0.0634814 0.0671458 0.320289

3 0.279175 0.0591380 0.147044 0.382800

4 0.331733 0.0559298 0.207031 0.429942

5 0.374484 0.0533648 0.255738 0.468376

6 0.410873 0.0512180 0.297124 0.501162

7 0.442778 0.0493668 0.333349 0.529970

8 0.471345 0.0477370 0.365730 0.555819

9 0.497327 0.0462798 0.395129 0.579377

10 0.521242 0.0449617 0.422145 0.601109

20 0.698954 0.0361002 0.621045 0.764443

30 0.827097 0.0312579 0.761380 0.885305

40 0.936591 0.0287349 0.878053 0.991814

50 1.03893 0.0281126 0.983610 1.09486

60 1.14127 0.0292822 1.08555 1.20152

70 1.25077 0.0322921 1.19109 1.31917

80 1.37891 0.0375327 1.31115 1.4603090 1.55662 0.0467137 1.47385 1.65984

91 1.58054 0.0480609 1.49552 1.68691

92 1.60652 0.0495471 1.51903 1.71637

93 1.63509 0.0512059 1.54482 1.74881

94 1.66699 0.0530861 1.57357 1.78509

95 1.70338 0.0552625 1.60629 1.82653

96 1.74613 0.0578580 1.64467 1.87530

97 1.79869 0.0610984 1.69175 1.93535

98 1.86856 0.0654773 1.75419 2.01532

99 1.97867 0.0725110 1.85234 2.14162

Interpretation: The goodness-of-fit tests (p-values = 0.909) suggest that the distributionand model fits the data adequately. The test for equal slopes is not significant (p-value =

0.673). The log ED50 for the four analgesics are 0.561374, 0.465283, 0.0123990 and

1.03893. ED50 values are 3.642286, 2.919329, 1.028961, 10.9378 . The relative potenciescan easily be worked out.

Documents

6-Statistical Methods in Bioefficicacy Trials