GATE:Statistical Inference in Pharmaceutical Sciences
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Introduction
Imagine yourself to be the M.D. of a pharmaceutical company. The chief of your R&D division comes to you with the proposal,
“Sir, we developed a most novel formulation of a drug, please introduce this into the market”. You have to take a decision and
this decision involves a risk. If you introduce it into the market and if it becomes a hit, you will be happy. But if it fails,
you lose money and you lose prestige as well. Suppose you hesitate to introduce it, fearing failure, your R&D chief may feel
disappointed with you, and resign. Then he may go to another company and give it to them. They may introduce it and earn profits.
In such a case, you would have lost a valuable opportunity to win profits and people may consider you a coward. So life involves
taking decisions, and in taking decisions you have to weigh benefits against risks and take a calculated risk. There may be other
situations demanding decisions; such as; 1. You are operating a tableting machine and based on a sample of tablets, the speed of
the machine has to be altered. 2. You have the option to use one among four methods of assay for a drug product and you have to
choose one. 3. You have to know whether the variation you are observing in the quality/drug content/weight of your emulsion
product is the same/less/more as the variation being observed by other manufacturers.
Like this, whatever may be your field of work, decisions have to be taken and you are often in a situation where you have to use
principles of statistical inference. .What we are doing is, we are applying the probability theory to our problems; this theory
can help us in solving problems in behavioral, natural, social, medical and pharmaceutical sciences. It provides an important
tool for the analysis of any situation which in some way involves an element of uncertainty or chance. Probability theory is the
basis for our methods, which we use when we generalize from observed data; these methods are methods of statistical
inference.
Before we go for the methods we must understand the terminology. Statistical data are the raw material of our investigations.
This data may be measured on four varieties of scales.
- Nominal data
- Ordinal data
- Interval data
- ratio data.
Nominal data
If the data can be divided into two or more categories based on qualitative differences and there is no gradation (like one is more/higher/nicer than the other) then the data are nominal data. Ex. 1. A class of students is divided into boys and girls 2. The drug products available are divided into tablets and capsules.
Ordinal data
When the data are divided into categories and the categories are arranged one above the other, we say that the data is an ordinal data.
Ex. 1. Bags of lactose obtained from different suppliers are categorized as excellent, good, average and poor 2. Materials used
in construction are categorized as strong, average, weak etc.
Interval data
If we can arrange the data on a scale, which has equal differences or intervals then we call it interval data. Ex. Temperature scale. Suppose we have different liquid samples at 1000C, 800C, 600C and 590C.
we can say that 1000C > 800C; 590C < 600C and 800C – 790C = 600C – 590C. But we cannot say that 80C water is twice as hot as 400C. This problem arises because the centigrade temperature scale has an artificial zero. If the temperature of a substance in zero OC, if does not indicate the absence of heat (which the temperature scale is trying to measure).
Ratio Scale
If we can also form quotients with the data, we call it ratio data. If a boy is 2ft. tall and if another boy is 4ft. tall, we can
say that the second boy is twice as tall as the first boy. Measurements of length, height, money amounts, weight, volume, area,
pressure, elapsed time, sound intensity, density, brightness, velocity etc. are all ratio scale measurements.
Population
The large unit or space or gathering that is under study is called a population. If a set of data consists of all conceivably possible (or hypothetically possible) observations of a certain phenomenon, we call it a population. If a set of data contains only a part of these observations, we call it a sample. A population may be real; we may be wishing to study the quality of empty large volume parenteral bottles [1 lakh bottles]; then our population is the 1 lakh empty bottles. To do this study we may pick 30 bottles at random and study them, then our sample size is 30. A population may be imaginary; suppose we are punching tablets and we are measuring the hardness of these tablets, then all the possible values that the hardness may take constitute an imaginary population.
Sample
A sample is a small subunit or a fraction of the large unit called population that we are studying. We study the sample and based on our studies we make conclusions regarding the population. From 1 lakh glass bottles [population] we may pick up 30 bottles at random and study them for their quality. Based on our observations of these 30 bottles, we form judgments regarding the entire 1 lakh bottles.
To distinguish between description of populations and description of samples, we use different words and symbols. When we talk about population characteristics we use the word parameter and when we talk about sample characteristics we use the word statistic. Population parameters are denoted by Greek letters, such as µ for mean and σ for standard deviation. Sample statistics are denoted by English letters, such as for mean and s for standard deviation.
Sampling Distribution
If we take several small samples from a large population; determine a statistic such as mean ( ) for all those samples and form a
frequency distribution with those mean ( ) values; then such a distribution is known as the sampling distribution and its
standard deviation is known as the standard error of the mean. For random samples of size n taken from a population having the
mean and the standard deviation σ , the theoretical sampling distribution of has the mean µ = µ and the standard deviation
depending on whether the population is infinite or finite of size N. is the standard error of the mean.
Standard Error
This standard error ( ) plays a very fundamental role in our statistical inference. It measures the extent to which sample means
can be expected to fluctuate, or vary, due to chance. The smaller the standard error the less the means are spread out and the
better are our chances that the estimate will be close. The standard error of the mean increases as the variability of the
population increases and it decreases as the sample size increases.
Testing of Hypothesis
Let us consider an imaginary situation. You have prepared two different drug products (a tablet and capsule) for a particular
drug. You want to determine whether one product is superior to another. So you take a sample of 10 tablets and find their effect
(bioavailability) on ten rats. One week later, you take a sample of 10 capsules and find their effect (bioavailability) on the
same ten rats. If you are measuring the bioavailability in terms of area under the curve; and if the difference observed is 20;
next you are faced with the question; is this 20 significant? Can we on the basis of this difference declare that one drug
product is superior to another? Is this difference due to a real reason (Is the quality of drug product superior)? Is this
difference merely due to chance reason (due to the chance reasons such as some small errors that may creep into the experimental
procedure unexpectedly). A real reason is a reason which is due to a built in difference or variability in the two drug products.
A chance reason has no basis and happens unexpectedly. However, systematic we are, we cannot completely replicate the conditions
of an experiment and some small stray or chance errors will creep in. However the effect of the real reason is considerable and
the effect of the chance reason is much smaller than the effect of the real reason. In testing of hypothesis we are testing to
see if the difference observed between two statistic values (two means or two variances) is merely due to chance (in which case
we declare it as “not significant” or due to a real reason (in which case we declare it as “significant”). In the first case the
ratio of is smaller than the ratio expected due to chance (given in tables). In the second case this ratio is larger than the
ratio expected due to chance. In making this decision we stand the risk of making two types of errors, Type I error and Type II
error.
Types of Errors
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decision from sample
reject H0 Accept Ho
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H0 true wrong, Type I error Correct
H0 false correct wrong Type II error
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In this testing of hypothesis we follow a particular procedure. If the null hypothesis of “no difference” is true and we accept
it, it is the correct decision, but if we reject if we are committing what is called as type I error or error of commission. If
the null hypothesis is false and if we reject it, we are taking the correct decision, but if we accept if we are committing what
is called as Type II error or error of commission. Taking again our example of “introducing a new formulation into the market;
the test and the analysis go like this: The R&D chief proposes to introduce a new formulation N.F into the market. He says
that it is much better than the old formulation O.F.
We take 100 units of N.F and study some property, such as bioavailability from these units in rats. Let us say the result is 150. One week later we take 100 units of O.F and study the same property in the same manner on the same rats and the result is 120. The difference observed is 30. Is this difference significant or not? We follow the testing procedure.
Case I : Suppose at 0.05 level of significance the test statistic is more than the table value; we reject the null hypothesis and declare the difference to be significant. Then we introduce the N.F into market. Suppose the experiment we did on a few rats with a few tablets or capsules is really indicative of the real nature of difference, then the people will find a real increase in bioavailability with our product, our N.F will be a hit and we will get profits and fame. Our decision here is proved correct. But suppose our experiment is not a true indicator of the real situation and suppose there is really not that much difference between O.F and N.F practically, then people will reject our N.F, it will become a flop; we lose money and prestige. We are committing here Type I error, rejecting a true HO. When we should not have taken any action, we are taking action, this is known as “error of commission”.
Case II : Suppose at 0.05 level of significance the test statistic is less than the table value; we accept the null
hypothesis and declare the difference to be not significant. Then we do not introduce the N.F. into the market. N.F. goes to a
rival manufacturer and he introduces it into the market. If our decision is right and there really is no difference between O.F
and N.F then we save a lot of money and time. And the rival manufacturer who introduced it into the market loses money, time and
prestige as well. But if our decision is wrong and it the N.F. becomes a hit, then the rival manufacturer gets profit and fame
and we are shown to be cowards, we lose face. Where we should have taken action we hesitated and stayed back, this is known as
error of omission or Type II error. So decision taking involves thinking on these terms, balancing the two errors, deciding which
error is more serious to us and taking a calculated risk. There is no decision which is absolutely correct and error free. There
are no rules which tell us which error should be minimized. But that test which can detect a difference that exists is known to
be more powerful. Probability of committing Type I error is called α and the probability of committing Type II error is called β.
(1-β) is known as the power of a test. Among the two errors, usually, error of omission (failure to see a difference which is
there) is considered more serious, and a test in which this error is less, is a more powerful test.
Significance Testing
When we conduct a statistical test to find out whether the difference observed between two statistic values or between a statistic and a parameter is significant or not it is called as significance testing. To do this we use some common probability distributions such as Z, t, χ 2 or F. The Z test, the t test and the F test are known as parametric tests. The χ 2 test comes under non-parametric tests. These tests concern the difference between a sample mean and a population mean, the difference between two sample means, the difference between a sample proportion and a population proportion and the difference between two sample proportions and the difference between two sample proportions. The F test concerns the difference between two variances.
Parametric Tests
Z test
: When the sample size is large, we perform the Z test. Usually if the sample size is less than 30 we consider it as small, if it is more than 30 we consider it as large. In performing this test we are usually comparing a population with a sample. In performing the test, we follow the same procedure for all the tests.
Example : A table manufacturer declares that in his diclofenac tablets the average drug content is 50mg per tablet, with a standard deviation of 2mg. A drugs inspector comes to the factory and picks up 50 diclofenac tablets, through a random sampling procedure, gets then analyzed and finds that the average drug content is 45mg per tablet. Now should the drugs inspector declare that the tablets are not really having the declared amount of drug?
The test is done in the following manner: 1. Null Hypothesis, HO : The average drug content of the tablets is 50mg. The difference observed between µ=50mg and =45 is not significant µ=5-mg. Alternative Hypothesis H1 : The difference observed is significant µ≠50mg. 2. α = 0.05 3. Criterion : At 0.05 level of significance the critical value of Z is ± 1.96 for a two tailed test. This is a two tailed test. 4. Formula : Z =
= sample mean = 45 = population mean = 45 = standard error of mean =
where = standard deviation of the population and n is the sample size.
Z =
Decision : At 0.05 level of significance, the critical value of Z = ± 1.96. The computed value of -17.85 is more than the critical value and falls in the rejection region. Thus we reject the HO and declare that the difference observed is significant. The tablets are not having an average drug content of 50mg. We can do this test at 0.01 level of significance also. In such as case, the critical value of Z is ± 2.58. Even in such a case the calculated value is more than the critical value, and we can reject HO. If we are considering only one side of a decision then we call it a one tailed test, if we are considering both sides of a test, we call it a two tailed test. Suppose we write H1, the alternative Hypothesis as µ < 50mg, the it becomes a one tailed test. That is, we are not interested in finding out whether μ > 50 or not, we are interested in only one side, i.e. whether the drug content is really 50mg or less than that. Then the critical value of Z will be ± 1.645 for 0.05 level of significance and ± 2.33 for 0.01 level of significance. Example : A manufacturer claims that his drug product has a mean shelf lie of 21.11 months with a standard deviation of 1.73 months. A random sample of 100 drug products shows a mean shelf life of 21.20 months. Test the validity of the manufacturer’s claim. 1. Null Hypothesis : The sample has been drawn from a population with a mean of 21.11 months and a standard deviation of 1.73 months. 2. α =0.05 3. Criterion : Z = ± 1.96 4. = 21.20 µ = 21.11 σ = 1.73 n = 100
5. Decision : The critical value of Z at 0.05 level of significance is ± 1.96. The calculated value of Z = 0.520 is less
than the table value of Z = ± 1.96. Thus we accept the validity of the claim of the manufacturer.The t-test
The tests of significance used for statistics of large samples are different from those which are used for small samples. This is because, when the sample size is large, we can make certain assumptions which are not true when the sample size is small. The assumptions are 1. The random sampling distribution of a statistic is approximately normal. 2. Values of the sample items are very close to the values of the population items and can be used in the place of the standard error of the estimate. For the Z test if the population standard deviation is not known we can make use of the sample standard deviation as an estimate. When the sample size is less than 30 we make use of the t test. Student t-Distribution : If we take a very large number of small samples from a population, calculate the mean for each sample and plot the frequency distribution of these means then the resulting sampling distribution is the student’s t-distribution.
The above diagram shows that the t-distribution is similar to a normal curve but a little flatter. The 95% limits of the
t-distribution will lie further from the mean in the t distribution than they do in the normal distribution. The t-distribution
curve changes with the sample size. As the sample size approaches 30, the t-distribution becomes more and more like the normal
curve.
t-tables : These tables contain t values for all possible sample sizes upto 30 and are given with reference to the degrees of freedom, and for one tailed test as well as for two tailed tests. They are also given at different levels of significance.
Degree of freedom : For the student’s t-distribution, the number of degrees of freedom is the sample size less one, i.e, (n-1) Formula : t = Where = sample mean µ = population mean S = standard error of mean and S = where S = sample standard deviation and n = sample size. The t test is done in different situations.
I. Comparing a sample mean with a population mean T = Where = sample mean O= population mean S = sample standard deviation n = sample size. Example : Suppose that we want to test, on the basis of a random sample of size n=5, whether or not the % of defective items in some batches of drug products exceeds 12 percent. If the sample has a mean % of defective items of 12.7 and if the standard deviation is 0.38 percent, what is your conclusion? Null Hypothesis : HO : μ = 12% H1 : μ >12% Level of significance : 2 = 0.01 Criterion : Reject HO if t > 3.747, the value of t at 0.01 level of significance and 5-1=4 degrees of freedom. Formula = t = Calculations : t = Decision : Since t = 4.12 exceeds 3.747 the null hypothesis HO must be rejected, the % of defective items exceeds 12%.
II. Difference between means : If the samples are large, we use the formula Z = Where and are the samples means and are the
sample standard deviations n1 and n2 are the sample sizes. If the sample sizes are small we use the formula T = Where and are
sample means S1 and S2 are sample standard deviations and n1 and n2 are sample sizes. Example : The following random samples
are measurements of the shelf life in hours of 5 batches of tablets of the same drug made from two different formulae. Use the
0.05 level of significance to test whether the difference between the means of these two samples is significant ______
Data : Formula I : 8400, 8230, 8380, 7869, 7930 Formula II : 7510, 7690, 7720, 8070, 7660 1. Null
Hypothesis : Alternative Hypothesis : 2. Level of Significance : = 0.05 3. Criterion : Reject the null
hypothesis if t<-2.306 or t>2.306 where t is given by the formula t = and 2.36 is the value of t 0.025 for 5+5-2=8 d.f;
otherwise state that the difference between the two sample means is not significant. 4. Calculations : Formula I :
8400, 8230, 8380, 7869, 7930 Formula II : 7510, 7690, 7720, 8070, 7660
= 8160 = 7730
S12 = 63, 450 S22 = 42, 650 So that = t = Decision : Since t=2.95 exceeds 2.306, the null hypothesis must be rejected, in other words, we conclude that the average shelf life in hours of the batches of tablets made from the two formula is not same, they are different.
Paired t test When the observations in the two samples are paired we perform a paired t test. Observations are considered paired when the two sets of measurements are taken on the same set of animals, before and after a treatment and in such other cases. Formula = t = Where d = X2 – X1 and is com
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