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8/4/2019 Ashwini Mpa Seminar
1/22
Click to edit Master subtitle style
4/8/12
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