Department of Community Medicine, BMC, Sagar

1. PARAMETRIC TESTS
Guided by- Presented
by-
Dr. Shraddha Mishra Dr. Shefali
Jain
(Associate Professor) (P.G. 2nd

2.  Let’s take an example to understand the concept of
Hypothesis Testing. A person is on trial for a criminal
offense and the judge needs to provide a verdict on his
case. Now, there are four possible combinations in such
a case:
 First Case: The person is innocent and the judge
identifies the person as innocent
 Second Case: The person is innocent and the judge
identifies the person as guilty
 Third Case: The person is guilty and the judge identifies
the person as innocent
 Fourth Case: The person is guilty and the judge
identifies the person as guilty

3.  As you can clearly see, there can be two types of error in
the judgment – Type 1 error, when the verdict is against
the person while he was innocent and Type 2 error, when
the verdict is in favor of Person while he was guilty
 According to the Presumption of Innocence, the person
is considered innocent until proven guilty. That means
the judge must find the evidence which convinces him
“beyond a reasonable doubt”. This phenomenon
of “Beyond a reasonable doubt” can be understood
as Probability (Judge Decided Guilty | Person is
Innocent) should be small

4.  The basic concepts of Hypothesis Testing are actually
quite analogous to this situation.
 We consider the Null Hypothesis to be true until we
find strong evidence against it. Then. we accept
the Alternate Hypothesis.
 We also determine the Significance Level (⍺) which can
be understood as the Probability of (Judge Decided
Guilty | Person is Innocent) in the previous example.
Thus, if ⍺ is smaller, it will require more evidence to
reject the Null Hypothesis.

6.  Directional Hypothesis
 In the Directional Hypothesis, the null hypothesis is
rejected if the test score is too large (for right-tailed and
too small for left tailed). Thus, the rejection region for
such a test consists of one part, which is right from the
center

7. Non-Directional Hypothesis
In a Non-Directional Hypothesis test, the Null Hypothesis is
rejected if the test score is either too small or too large. Thus,
the rejection region for such a test consists of two parts: one
on the left and one on the right

8.  p-value has the benefit that we only need one value to
make a decision about the hypothesis. We don’t need to
compute two different values like critical value and test
scores. Another benefit of using p-value is that we can
test at any desired level of significance by comparing
this directly with the significance level

9.  Critical Value is the cut off value between Acceptance
Zone and Rejection Zone. We compare our test score to
the critical value and if the test score is greater than the
critical value, that means our test score lies in the
Rejection Zone and we reject the Null Hypothesis. On
the opposite side, if the test score is less than the Critical
Value, that means the test score lies in the Acceptance
Zone and we fail to reject the null Hypothesis

10. Statistical Test
 These are intended to decide whether a hypothesis about
distribution of one or more populations should be
rejected or accepted.
 These may be:
 Parametric Test
 Non Parametric Test

11.  These tests the statistical significance of the:-
1) Difference in sample and population means.
2) Difference in two sample means
3) Several population means
4) Difference in proportions between sample and
population
5) Difference in proportions between two independent
populations
6) Significance of association between two variables

12. System for statistical Analysis
1. State the Research Hypothesis
2. State the Level of Significance
3. Calculate the test statistic
4. Compare the calculated test statistic with the tabulated
values
5. Decision
6. Statement of Result

13. when to use parametric test?
 To use parametric test the following conditions have to
be satisfied:
 Data must be either in Interval scale or ratio scale.
 Subjects should be randomly selected.
 Data should be normally distributed

14. Determination of parametric test
 Interval scale: interval between observation in terms of
fixed unit of measurement. Eg. Measures of
temperature.
 Ratio scale: The scale has a fundamental zero point.
Eg. Age, income. (IN CASE OF NOMINAL AND ORDINAL
SCALE NON- PARAMETRIC TEST IS USED)

15. Types of parametric tests
 Large sample tests. - Z test
 Small sample test - t-test
 Independent /unpaired – t test
 Paired t-test
• ANOVA(analysis of variance)
 one way analysis of variance
 Two way analysis of variance

16. STUDENT’S T-TEST
 Developed by Prof W.S Gossett in 1908, who published
statistical papers under the pen name of ‘Student’. Thus
the test is known as Student’s ‘t’ test.
 Indications for the test:-
1. When samples are small
2. Population variance are not known.

17.  Uses
1. Two means of small independent samples
2. Sample mean and population mean
3. Two proportions of small independent samples

18.  Assumptions made in the use of ‘t’ test
1. Samples are randomly selected
2. Data utilised is Quantitative
3. Variable follow normal distribution
4. Sample variances are mostly same in both the groups
under the study
5. Samples are small, mostly lower than 30

19. ONE SAMPLE T-TEST
 When compare the mean of a single group of
observations with a specified value
 In one sample t-test, we know the population mean. We
draw a random sample from the population and then
compare the sample mean with the population mean and
make a statistical decision as to whether or not the
sample mean is different from the population

22.  Now we compare calculated value with table value at
certain level of significance (generally 5% or 1%)
 If absolute value of ‘t’ obtained is greater than table
value then reject the null hypothesis and if it is less than
table value, the null hypothesis may be accepted

23. Two Sample ‘t’ test
A. Unpaired Two sample ‘t’- test
 Unpaired t- test is used when we wish to compare two
means
 Used when the two independent random samples come
from the normal populations having unknown or same
variance
 We test the null hypothesis, that the two population
means are same i.e µ1= µ2 against an appropriate one
sided or two sided alternative hypothesis

24.  Assumptions:
 The samples are random & independent of each other
 The distribution of dependent variable is normal.
 The variances are equal in both the groups

28.  PAIRED TWO-SAMPLES T-TEST
 Used when we have paired data of observations from
one sample only, when each individual gives a pair of
observations.
 Same individuals are studied more than once in different
circumstances- measurements made on the same people
before and after interventions

29.  Assumptions:
 The outcome variable should be continuous
 The difference between pre-post measurements should
be normally distributed
• Instead of using a series of individual comparisons we
examine the differences among the groups through an
analysis that considers the variation among all groups at
once. i.e. ANALYSIS OF VARIANCE

30. Analysis of Variance(ANOVA)
 Given by Sir Ronald Fisher
 The principle aim of statistical models is to explain the
variation in measurements.
 Analysis of variance (ANOVA) is a statistical technique
that is used to check if the means of two or more groups
are significantly different from each other. ANOVA
checks the impact of one or more factors by comparing
the means of different samples.

31.  Another measure to compare the samples is called a t-
test. When we have only two samples, t-test and
ANOVA give the same results. However, using a t-test
would not be reliable in cases where there are more than
2 samples. If we conduct multiple t-tests for comparing
more than two samples, it will have a compounded
effect on the error rate of the result.
 Assumptions for ANOVA
 Sample population can be easily approximated to normal
distribution.
 All populations have same Standard Deviation.
 Individuals in population are selected randomly.
 Independent samples

32.  The Null hypothesis in ANOVA is valid when all the
sample means are equal, or they don’t have any
significant difference. Thus, they can be considered as a
part of a larger set of the population. On the other hand,
the alternate hypothesis is valid when at least one of the
sample means is different from the rest of the sample
means.

34.  ANOVA compares variance by means of a simple
ratio, called F-Ratio
 F=Variance between groups/Variance within groups
 The resulting F statistics is then compared with
critical value of F (critic), obtained from F tables in
much the same way as was done with ‘t’
 If the calculated value exceeds the critical value for
the appropriate level of α, the null hypothesis will
be rejected.

35. Variance between groups

36. F- statistics
F=Variance between groups/Variance within groups

38.  A F test is therefore a test of the Ratio of Variances
 F Tests can also be used on their own, independently of
the ANOVA technique, to test hypothesis about
variances.
 In ANOVA, the F test is used to establish whether a
statistically significant difference exists in the data being
tested.
ANOVA
ONEWAY
TWO WAY

39. One Way ANOVA
 If the various experimental groups differ in terms of
only one factor at a time- a one way ANOVA is used
 e.g. A study to assess the effectiveness of four different
antibiotics on S Sanguis
 E.g. You have a group of individuals randomly split into
smaller groups and completing different tasks. For
example, you might be studying the effects of tea on
weight loss and form three groups: green tea, black tea,
and no tea.

40. Two Way ANOVA
 If the various groups differ in terms of two or more
factors at a time, then a Two Way ANOVA is performed
 e.g. A study to assess the effectiveness of four different
antibiotics on S Sanguis in three different age groups
 E.g. you might want to find out if there is an interaction
between income and gender for anxiety level at job
interviews. The anxiety level is the outcome, or the
variable that can be measured. Gender and Income are
the two categorical variables.

41. Pearson’s Correlation Coefficient
 Correlation is a technique for investigating the
relationship between two quantitative, continuous
variables
 Pearson’s Correlation Coefficient(r) is a measure of
the strength of the association between the two
variables.

42.  Assumptions Made in Calculation of ‘r’
1. Subjects selected for study with pair of values of X & Y
are chosen with random sampling procedure.
2. Both X & Y variables are continuous
3. Both variables X & Y are assumed to follow normal
distribution

43. Steps
 The first step in studying the relationship between two
continuous variables is to draw a scatter plot of the
variables to check for linearity.
 The correlation coefficient should not be calculated of
the relationship is not linear
 For correlation only purposes, it does not matter on
which axis the variables are plotted

44.  However, conventionally, the independent variable is
plotted on X axis and dependent variable on Y-axis
 The nearer the scatter of points is to a straight line, the
higher the strength of association between the variables.

45. Types of Correlation
 Perfect Positive Correlation r=+1
 Partial Positive Correlation 0<r<+1
 Perfect negative correlation r=-1
 Partial negative correlation 0>r>-1
 No Correlation

47. Z Test
 z tests are a statistical way of testing a hypothesis when
either:
 We know the population variance, or
 We do not know the population variance but our sample
size is large n ≥ 30
 If we have a sample size of less than 30 and do not know
the population variance, then we must use a t-test

48.  Assumptions to apply Z test
 The sample must be randomly selected Data must be
quantitative
 Samples should be larger than 30
 Data should follow normal distribution
 Sample variances should be almost the same in both the
groups of study

50.  If the SD of the populations is known, a Z test can be
applied even if the sample is smaller than 30

51. Indications for Z Test
 To compare sample mean with population mean
 To compare two sample means
 To compare sample proportion with population
proportion
 To compare two sample proportions

52. Steps
1. Define the problem
2. State the null hypothesis (H0) & alternate hypothesis
(H1)
3. Find Z value
Z= Observed mean-Mean Standard Error
4. Fix the level of significance
5. Compare calculated Z value with the value in Z table at
corresponding degree significance level.
 If the observed Z value is greater than theoritical Z
value, Z is significant, reject null hypothesis and accept
alternate hypothesis

55.  One tailed and Two tailed Z tests •
 Z values on each side of mean are calculated as +Z or as
-Z.
 A result larger than difference between sample mean
will give +Z and result smaller than the difference
between mean will give -Z

56. Two sample Z test

59.  E.g. for two tailed: In a test of significance, when one
wants to determine whether the mean IQ of
malnourished children is different from that of well
nourished and does not specify higher or lower, the P
value of an experiment group includes both sides of
extreme results at both ends of scale, and the test is
called two tailed test.
 E.g. for single tailed: In a test of significance when one
wants to know specifically whether a result is larger or
smaller than what occur by chance, the significant level
or P value will apply to relative end only e.g. if we want
to know if the malnourished have lesser mean IQ than
the well nourished, the result will lie at one end ( tail )of
the distribution, and the test is called single tailed test

60. Conclusion
 Tests of significance play an important role in
conveying the results of any research & thus the choice
of an appropriate statistical test is very important as it
decides the fate of outcome of the study.
 Hence the emphasis placed on tests of significance in
clinical research must be tempered with an
understanding that they are tools for analyzing data &
should never be used as a substitute for knowledgeable
interpretation of outcomes.