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SEMINAR ON

STUDENT`S T

TEST

BY

K.ASHWINI

I MPHARMA

DEPT OF INDUSTRIAL PHARMACY

UNDER THE GUIDANCE OF

PROF. A CENDIL KUMAR


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A t test is any statistical hypothesis test in which the

test statistic follows a student`s t distribution, if the null hypothesis

is supported.

It is most commonly applied when the test statistic

would follow a normal distribution, if the value of a scaling term

in the test statistic were known. When the scaling term is unknown& replaced by an estimated based on the data, the test

statistic( under certain conditions) follows a student t distribution.

DEFINITION


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This is one of the most widely used tests in pharmacological

investigation involving the use of small samples. The ttest is

applied for analysis when the number of samples is about 30 or less.

It is usually applicable for analysis applicable to measurement

(graded) data, such as blood sugar level, body weight, reaction time,

etc. it can be used under two situations

Student`s t test for small samples:


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1) When the comparison is made between two measurements in

two different groups (between subject comparison), and

2) When the comparison is made between two measurement in

the same subject (within subject comparison by paired t test)

following two consecutive treatments provide the first has no

influence on the effect of the second.


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ANDDATA

OFT-TEST

N

DF

Compareonegroup toahypothetical value

One samplettest

Subjects arerandomlydrawn froma populationanddistributionof the meanbeing testedis normal

Usuallyused tocomparethe meanof a sampleto a knownnumber(often0)

n-1

Compare

two

Unpair

edt

Two -sample

assuming

Two

samples


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two sampleassumingunequalvariance(heteroscedas

tic t test

The variance inthe two groupsare extremelydifferent. e.g.the two

samples are ofvery differentsizes

Compare

twopairgroups

Paired t

test

The observeddata are fromthe samesubject or froma matchedsubject and are

drawn from apopulationwith a normaldistributiondoes notassume that

the variance ofboth

used to comparemeans on thesame or relatedsubject over timeor in differingcircumstances;

subjects areoften tested in abefore-aftersituation

n-1


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Independent one-samplet-test

The One-Sample T Test compares the mean score of a sample to a

known value. Usually, the known value is a population mean. In

testing the null hypothesis that the population mean is equal to a

specified value0, one uses the statistic

where

sis the standard deviation of the sample and

X = Sample mean, 0 = population mean

n = number of observations in sample

The degrees of freedom used in this test isn 1.


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TWO PAIRED SAMPLES: WITHIN-SUBJECT

DESIGNS

-Hypothesis test

-Confidence Interval

-Effect Size TWO INDEPENDENT SAMPLES: BETWEEN-SUBJECT

DESIGNS

-

Hypothesis test- Confidence interval

- Effect Size


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TWO KINDS OF STUDIES

There are two general research strategies that can beused to obtain the two sets of data to be compared:

1. The two sets of data could come from two independentpopulations (e.g. women and men, or students from section A and

from section B)


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Two sample t tests:

Two-sample t-tests for a difference in mean can be

either unpaired or paired.

Paired t-tests are a form of blocking, and have

greater power than unpaired tests, when the paired units are similar

with respect to "noise factors" that are independent of membership inthe two groups being compared. In a different context, paired t-tests

can be used to reduce the effects of confounding factors in an

observational study.


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The unpaired, or "independent samples"t-test is used when two

separate sets of independent and identically distributed samples are

obtained, one from each of the two populations being compared.

For example, suppose we are evaluating the effect of a medical

treatment, and we enroll 100 subjects into our study, then randomize 50

subjects to the treatment group and 50 subjects to the control group. In this

case, we have two independent samples and would use the unpaired form of

thet-test. The randomization is not essential hereif we contacted 100

people by phone and obtained each person's age and gender, and then used a

two-sample t-test to see whether the mean ages differ by gender, this would

also be an independent samplest-test, even though the data are


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Dependent samples (or "paired")t-teststypically consist

of a sample of matched pairs of similarunits, or one group of units

that has been tested twice (a "repeated measures"t-test).

A typical example of the repeated measurest-test would be

where subjects are tested prior to a treatment, say for high blood

pressure, and the same subjects are tested again after treatment with a

blood-pressure lowering medication.

A dependentt-test based on a "matched-pairs sample" results from

an unpaired sample that is subsequently used to form a paired

sample, by using additional variables that were measured along with


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The matching is carried out by identifying pairs of

values consisting of one observation from each of the two samples,

where the pair is similar in terms of other measured variables. This

approach is often used in observational studies to reduce or eliminate

the effects of confounding factors.


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INDEPENDENT TWO-SAMPLET-TEST

1) Equal sample sizes, equal variance:

This test is only used when boththe two sample sizes (that

is, the number,n, of participants of each group) are equal;

It can be assumed that the two distributions have the same variance.

Violations of these assumptions are discussed below.

Thetstatistic to test whether the means are different can be

calculated as follows:


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Where

Here,

is the grand standard deviation(orpooled standard

deviation), 1 = group one, 2 = group two.

The denominator oftis thestandard errorof the difference between

two means.

For significance testing, thedegrees of freedomfor this test is

2n 2


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Unequal sample sizes, equal variance:

This test is used only when it can be assumed that the two

distributions have the same variance. Thetstatistic to test whether

the means are different can be calculated as follows:

Where,


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Here

is an estimator of the common standard deviation of the

two samples: it is defined in this way so that its square is an

unbiased estimatorof the common variance whether or not the

population means are the same.

In these formulae,

n= number of participants, 1 = group one, 2 = group two.

n 1 is the number of degrees of freedom for either group,

and the total sample size minus two (that is,n1+n2 2) is the total

number of degrees of freedom, which is used in significance testing.


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3)Unequal sample sizes, unequal variance

This test is used only when the two population variances are

assumed to be different (the two sample sizes may or may not be

equal) and hence must be estimated separately. Thetstatistic to

test whether the population means are different can be calculated as

follows:

where


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Wheres2is the unbiased estimatorof the varianceof the two

samples,

n= number of participants, 1 = group one, 2 = group two.

Note that in this case, is not a pooled variance.

For use in significance testing, the distribution of the test statistic is

approximated as being an ordinarystudent`s t distribution with the

degrees of freedom calculated using


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DEPENDENT"t-TEST FOR PAIRED SAMPLES

This test is used when the samples are dependent; that is, when

there is only one sample that has been tested twice (repeated

measures) or when there are two samples that have been matched or

"paired". This is an example of a paired difference test.


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From this equation,

The differences between all pairs must be calculated. The pairs

are either one person's pre-test and post-test scores or between pairs

of persons matched into meaningful groups (for instance drawn from

the same family or age group: see table).

The average (XD) and standard deviation (sD) of those

differences are used in the equation.

The constant0is non-zero if you want to test whether the average

of the difference is significantly different from0.

The degree of freedom used isn 1.


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EXAMPLE: Students t-test

The Students t-test compares the averages and standard deviations

of two samples to see if there is a significant difference between

them.

We start by calculating a number, t

tcan be calculated using the equation

Where: x1 is the mean of sample 1

s1 is the standard deviation of sample 1

n1 is the number of individuals in sample 1

x2 is the mean of sample 2

s2 is the standard deviation of sample 2

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