The document discusses various parametric statistical tests including t-tests, ANOVA, ANCOVA, and MANOVA. It provides definitions and assumptions for parametric tests and explains how they can be used to analyze quantitative data that follows a normal distribution. Specific parametric tests covered in detail include the independent samples t-test, paired t-test, one-way ANOVA, two-way ANOVA, and ANCOVA. Examples are provided to illustrate how each test is conducted and how results are interpreted.

2. Parametric Test :
t2 test
anova
ancova
manova
Princy Francis M
Ist Yr MSc(N)
JMCON

3. DEFINITION
Statistics is a branch of science that deals with the
collection, organisation, analysis of data and drawing of
inferences from the samples to the whole population.
Statistical tests are intended to decide whether a
hypothesis about distribution of one or more
populations or samples should be rejected or accepted.

4. DEFINITION
Inferential statistics is the statistics that permit
inferences on whether the results observed in a sample
are likely to occur in the larger population
Parametric test is a class of statistical tests that involve
assumptions about the distribution of the variables and
estimation of a parameter.

5. Parametric test
• Parametric test is a statistical test that makes assumptions about
the parameters of the population distribution(s) from which one’s
data is drawn.
• most commonly used.
• the findings are inferred to the parameters of a normally
distributed populations.
• Numerical data (quantitative variables) that are normally
distributed are analysed with parametric tests.
• Parametric tests are done on the basis of mean and standard
deviation

6. ASSUMPTIONS
It requires 3 assumptions for using :
• the sample was drawn from a population for which the
variance can be calculated. It is expected to be in normal
distribution.
• the levels of measurement should be atleast interval
data or ordinal data with an approximately normal
distribution.
• the data can be treated as random samples.

7. Application of parametric test
• Used for Quantitative data.
• Used for continuous variables.
• Used when data are measured on approximate interval or ratio scales
of measurement.
• Data should follow normal distribution.

8. PARAMETRIC
STATISTICAL
ANALYSIS
Student's t-test
 Analysis of variance (ANOVA)
Analysis of Covariance (ANCOVA)
Multivariate analysis of variance
(MANOVA)

9. Student's t-test
• Developed by Prof.W.S.Gossett
• Student's t-test is used to test the null hypothesis that
there is no difference between the means of the two
groups
• Indication for t test
– When samples are small (<30)
– Population variance are not known

10. TYPES
One-sample t-test
Independent Two Sample T
Test (the unpaired t-test)
The paired t-test

11. One-sample t-test
• To test if a sample mean (as an estimate of a population mean) differs
significantly from a given population mean.
• The mean of one sample is compared with population mean
where 𝑥 = sample mean, u = population mean and S = standard
deviation, n = sample size

12. Example
A random sample of size 20 from a normal population gives a sample
mean of 40, standard deviation of 6. Test the hypothesis is population
mean is 44. Check whether there is any difference between mean.
• H0: There is no significant difference between sample mean and
population mean
• H1: There is significant difference between sample mean and
population mean
mean = 40 , 𝜇 = 44, n = 20 and S = 6

13. • tcalculated = 2.981
• t table value = 2.093
• tcalculated > t table value ;
Reject H0.

14. Independent Two Sample T Test (the
unpaired t-test)
• To test if the population means estimated by two independent
samples differ significantly.
• Two different samples with same mean at initial point and compare
mean at the end

15. t =
𝑥1− 𝑥2
𝑛1−1 𝑆1
2+ 𝑛2−1 𝑆2
2
𝑛1+𝑛2−2
1
𝑛1
+
1
𝑛2
Where 𝑥1 - 𝑥2 is the difference between the means of the two groups
and S denotes the standard deviation.
Example :
• Compare the height of girls and boys.
• compare 2 stress reduction interventions

16. ILLUSTRATION
Mean Hb level of 5 male are 10, 11, 12.5, 10.5, 12 and 5 female are 10,
17.5, 14.2,15 and 14.1 . Test whether there is any significant difference
between Hb values.
• H0: There is no significant difference between Hb Level
• H1: There is significant difference between Hb level.
t =
𝑥1− 𝑥2
𝑛1−1 𝑆1
2+ 𝑛2−1 𝑆2
2
𝑛1+𝑛2−2
1
𝑛1
+
1
𝑛2

17. • 𝑥1 = 11.2 , 𝑥2 =14.16 , 𝑆1
2
= 1.075, 𝑆2
2
= 7.293
• tcalculated = 2.287, t table = 2.306, tcalculated > t table value ; reject H0.
X1 X2 X1 - 𝑥1 X2 - 𝑥2 (X1 - 𝑥1)2 (X2 - 𝑥2)2
10
11
12.5
10.5
12
10
17.5
14.2
15
14.1
-1.2
- 0.2
1.3
-0.7
0.8
-4.16
3.34
0.04
0.84
-0.06
1.44
0.04
1.69
0.49
0.64
17.305
11.156
0.0016
0.706
0.0036
Σ = 56 70.8 4.3 29.172

18. The paired t-test
• To test if the population means estimated by two dependent samples differ
significantly .
• A usual setting for paired t-test is when measurements are made on the
same subjects before and after a treatment.
where 𝑑 is the mean difference and Sd denotes the standard deviation of the
difference.

19. Example
Systolic BP of 5 patients before and after a drug therapy is
Before 160, 150, 170, 130, 140
After 140, 110, 120, 140, 130
Test whether there is any significant difference between BP level.
• H0: There is no significant difference between BP Level before and after
drug
• H1: There is significant difference between BP level before and after drug

20. • 𝑑 = 22, Sd = 23.875
• tcalculated = 2.060, t table = 2.567, tcalculated < t table value ; Accept H0.
Before After d d- 𝑑 (d- 𝑑 )2
160
150
170
130
140
140
110
120
140
130
20
40
50
-10
10
-2
18
28
-32
-12
4
324
784
1024
144
𝛴𝑑 = 110 2280

21. ANALYSIS OF VARIANCE (ANOVA)
• R. A. Fischer.
• The Student's t-test cannot be used for comparison of three or more groups.
• The purpose of ANOVA is to test if there is any significant difference between the
means of two or more groups.
• The analysis of variance is the systematic algebraic procedure of decomposing the
overall variation in the responses observed in an experiment into variation.
• Two variances – (a) between-group variability and (b) within-group variability that
is variation existing between the samples and variations existing within the
sample.
• The within-group variability (error variance) is the variation that cannot be
accounted for in the study design.
• The between-group (or effect variance) is the result of treatment

22. ASSUMPTIONS OF ANOVA
• The population in which samples are drawn should be normal
• The sample observations are independent of each other
• The samples are selected at random
• The samples are drawn from population having equal variance
• The sample size should not differ widely
• The various effects(treatment and errors) are additive in nature
• The experimental error are normally and independently distributed
with mean Zero

23. • A simplified formula for the F statistic is
where MST is the mean squares between the groups and MSE is the
mean squares within groups

24. TYPES  ONE WAY ANOVA
 TWO WAY ANOVA

25. ONE WAY ANOVA
• It compares three or more unmatched groups when data
are categorized in one way.
Total sum of Square(TSS) = treatment sum of square(SST)
+ error sum of square(SSE)
Example.
• 1. Compare control group with three different doses of
aspirin in rats
• 2. Effect of supplementation of vit C in each subject
before, during and after the treatment.

26. One way ANOVA table
Source d.f Sum of
Square
Mean Square F
Between
treatment
Within
treatment
error
t-1
n-t
SST
SSE
MST = SST/t-1
MSE = SSE/n-t
F= MST/MSE
~ F (t-1),(n-t)
Total n-1 TSS

27. TWOWAY ANOVA
• It is used to determine the effect of two nominal predictor
variables on a continuous outcome variable.
• A two-way ANOVA test analyzes the effect of the independent
variables on the expected outcome along with their
relationship to the outcome itself.
Example :
• Effect of two antihypertensive drugs in two different doses
• Comparing the employee productivity based on the working
hours and working conditions

28. Two way ANOVA table
Source d.f Sum of
Square
Mean Square F
Treatme
nt
Blocks
Error
t-1
r-1
(r-1)(t-1)
SST
SSB
SSE
MST = SST/t-1
MSB = SSB/r-1
MSE = SSE/(r-1)(t-
1)
Ft= MST/MSE
~ F (t-1), (t-1) (r-1)
Fb= MSB/MSE
~ F (r-1), (t-1) (r-1)
Total rt-1 TSS

29. Difference between one & two way ANOVA
• a one-way ANOVA is used to determine if there is a difference in the
mean height of stalks of three different types of seeds.
• Since only 1 factor that could be making the heights different.
• if three different types of seeds, and then add the possibility that
three different types of fertilizer is used
• The mean height of the stalks could be different for a combination of
several reasons.
• Two factors (type of seed and type of fertilizer), use a two-way
ANOVA.

30. ANALYSIS OF COVARIANCE (ANCOVA)
ACOVA is a technique that combines the feature of analysis of
variance and regression.
It is used to increase the precision of treatment comparisons.
This method is based on the fact that there are some extraneous
sources of variation which also contribute to the experimental error
but are not controlled. These additional variations are known as the
ancillary or concomitant variates.

31. DEFINITION
• The very logical procedure, which reduces the experimental error
by eliminating from it the effects of variations in the concomitant
variate and thus increase the precision of the main variate on the
concomitant variate is known as Analysis of Covariance
(ANCOVA)

32. Example:
 Effect of 3 diet on gaining weight of animal
with different age group and different
initial weight will influence animal
performance and precision of experiment.

33. Assumptions
All assumptions of ANOVA is applicable here too. In
addition, it is assumed that,
• The relationship between X and Y is linear
• The relationship is same for each treatment
• The covariates are not affected by treatment
• The observations are from normal populations

34. USES
◦ To increase the precision in a randomized experiment
◦ To remove the effects of disturbing variables in
observational studies
◦ To throw light on the nature of treatment effects
◦ To analyse the data when some observations are missing
◦ To fit regression in multiple classification

35. WORKING OF ANCOVA
• ANCOVA works by adjusting the total SS, group SS, and error SS of the
independent variable to remove the influence of the covariate
Yij = µ + ti + β (xij - 𝑥) + eij
• Where Yij is the jth observation on the response variable under ith
treatment , µ is mean of x variable, ti is ith treatment effect
• β linear regression coefficient
• eij random error component which is independently and normally
distributed with mean zero and variance

36. Sum of Squares

37. Sum of Products
• To control for the covariate, the sum of products (SP) for the
dependent variable and covariate must also be used.
•
• x is the covariate, and y is the dependent variable. i is the individual
subject, and j is the group.

38. Error Sum of Products
• This is the sum of the products of the dependent variable and
residual minus the group means of the dependent variable and
residual.

39. Adjusting the Sum of Squares
• Using the SS’s for the covariate and the dependent variable, and the
SP’s, adjust the SS’s for the dependent variable

40. Test Statistic
• F = Mean square of β / adjusted error mean square

41. ADVANTAGES
• Better power
• Improved ability to detect and estimate
interactions
• The availability of extensions to deal with
measurement error in the covariates.

42. DISADVANTAGES
• There will be cost of introducing the blocking factor.
• It may be difficult to find blocking factors that are highly
correlated with the dependent variable.
• Loss of power may occur if a poorly correlated blocking
factor is used.

43. MULTIVARIATE ANALYSIS OFVARIANCE
(MANOVA)
• variation of ANOVA.
• MANOVA assesses the statistical significance of the effect of one
or more independent variables on a set of two or more
dependent variables.
• MANOVA has the ability to examine more than one dependent
variable at once or simultaneous effect of independent variables
on multiple dependent variables.
• ControlType I error.

44. EXAMPLE
 Case study on 2 different text books & student improvement in
maths and physics.
 Dependent variables here are improvement in maths and physics.
 Hypothesis: both the Dependent Variable are affected by difference
in text books.

45. ASSUMPTIONS
Multivariate normality :
 dependent variables should be normally distributed
within groups.
 Linear combinations of dependent variables must be
distributed.
 All subjects of variables must have multivariate normal
distribution.
Homogeneity of covariance matrices.
 The inter correlations (co variances) of the multiple
dependent variable across the cells of design.

46. ASSUMPTIONS
Independence of observations
 Subject score on dependent variables are not
influenced or related to other subject scores.
 Linearity
 Linear relationship against all pairs of dependent
variables, all pairs of covariates, all dependent
variable – covariate pairs in each cell.
 Therefore if relationship deviates from linearity
the power of analysis will be compromised.

47. ADVANTAGES
• MANOVA enables to test multiple dependent variables.
• MANOVA can protect against Type I errors

48. DISADVANTAGES
• MANOVA is many times more complicated than ANOVA, making it a
challenge to see which independent variables are affecting dependent
variables.
• One degree of freedom is lost with the addition of each new variable.

49. SUMMARY OF PARAMETRIC TESTS APPLIED FOR
DIFFERENT TYPE OF DATA
Type of Group Parametric test
Comparison of two paired groups Paired t test
Comparison of two unpaired groups Unpaired two sample t
test
Comparison of population and sample
drawn from the same population
One sample t test
Comparison of 3 or more matched groups
but varied in 1 factors
One Way Anova
Comparison of 3 or more matched groups
but varied in 2 factor
Two Way Anova

50. JOURNAL ABSTRACT
A Multivariate Analysis (MANOVA) of Where Adult
Learners Are In Higher Education
American institutions of higher education were originally established
with the purpose of educating the advantaged youth.
Due to this increase in adults re-entering the academy, it is
appropriate and timely to ask where these students are attending
school, what is known about their distribution in the higher
education system, and whether they are assembled in one type of
institution or evenly distributed among institutions.

51. Therefore, the purpose of this study was to determine where
undergraduate adult students are located within the 4-year private,
public, and for-profit universities offering undergraduate degrees in the
United States.
This study utilized descriptive and multivariate analyses of variance
(MANOVA) statistical analyses.
Descriptive analysis provided the number, means, and standard
deviations for college and university enrolments obtained from the
Integrated Postsecondary Education Data System (IPEDS) of the
National Center for Education Statistics (NCES) to answer two research
questions.
Two MANOVAs and comparative designs were employed to examine
electronic data accessed through IPEDS. Undergraduate students under
the age of 25 are enrolling in 4-year public and private universities in
the United States at about double the enrolment rate as that of for-profit
universities.

52. Using ANOVA to Examine the Relationship
between Safety & Security and Human
Development
This study aimed to examine the relationship between safety and
security index and human development.
The sample consisted of 53 African countries. The research question
is Does a statistical significant relationship exist between safety and
security index and human development.
In the process of examining the relationship between variables,
researchers can use t test or ANOVA to compare the means of two
groups on the dependent variable.

53. The main difference between t-test and ANOVA is that t test can only be
used to compare two groups while ANOVA can be used to compare two
or more groups.
In the process of selecting the data analysis technique for this study,
considered both ANOVA and t-test.
The advantage ANOVA has over t-test is that the post-hoc tests of
ANOVA allow to better controlling type 1 error.
Therefore, in order to control type 1 error, chose ANOVA as data
analysis technique for this study.
The results indicated that there is a statistically significant relationship
with strong effect size between safety and security index and human
development.

54. ASSIGNMENT
•WRITE AN ASSIGNMENT ON PARAMETRIC STATISTICAL APPLICATION
IN NURSING.

55. REFERENCES
• Mahajan BK. Methods in Biostatistics. Sixth edition. New Delhi: Jaypee
brothers medical publishers. 2003
• Rao SSSP. Biostatistics. Third edition. New Delhi: Prentice Hall India Pvt
Ltd;2004.
• Agarwal LB. Basic Biostatistics. Fourth edition. India: Newage international
publishers:2006
• Ali Z, Bhaskar BS. Basic Statistical Tools in research and data analysis. Indian
J Anaesth. 2016 Sep; 60(9): 662–669.
• Sundaram RK, Dwivedi SN, Sreenivas V. Medical Statistics : Principles and
methods. Second edition. New Delhi: Wolter Kluwer publication; 2015.