Tests of Significance.pptx powerpoint presentation

Presenter: Dr Vidya D C
Guide: Ms Radhika K
Dr N S Murthy
Acknowledgement: Mr Shivraj N S
Tests of Significance

10/16/2024
Contents
 Introduction
 Terms and concepts in test of significance
 Methods of tests of significance
 Steps in tests of significance
 Parametric methods
 Non parametric methods
2

10/16/2024
 Clinical significance v/s statistical significance
 Summary
 Conclusion
 References
3
Contents

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Introduction
4
 Statistics – methodology of collection and
meaningful interpretation of the data
Statistics
Descriptive statistics Inferential statistics

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Introduction
5
 Tests of significance – methodologies of statistics,
which deals with the techniques to analyse, how
far the difference between estimates from different
sample are due to sampling variation or otherwise.

10/16/2024
Terms and concepts in test of significance
6
 Null Hypothesis- first step in testing of
hypothesis.
 Level of significance –probability of committing
the type I error ( fixed at 5% or 1%)
 Parameter: It is a population value or a variable
that describes the population

10/16/2024
7
 Large sample: Refers to samples having >30
observations.
 Small sample: Refers to samples having ≤30
observations.
 Critical regions- Region of acceptance & Region of
rejection
Terms and concepts in test of significance

10/16/2024
Methods of tests of significance
8

PARAMETRIC TESTS- Assumptions
 The observations must be drawn from normally
distributed populations
 These populations must have the same variances
 The observations must be independent
 Population parameters involved is mean, standard
deviation
 Require interval scale or ratio scale (whole numbers
or fractions). Example: Height in inches: 72, 60.5,
54.7; temperature[30-34 degree Celsius]
9

PARAMETRIC TESTS…
 Critical ratio/ Z-test
 Paired t-test
 Unpaired t-test
 One way ANOVA
 Two way ANOVA
10

Steps in tests for large sample
1)Set up the null hypothesis that the 2 samples are
from the same population and that the difference
between the 2 sample estimates is due to sampling
variation.
2)Calculate 2 means x1 and x2 or 2 proportions p1
and p2 corresponding to the 2 samples with sample
size n1 and n2 respectively.
3)Calculate the standard deviation of the 2 samples
and their standard errors (SE1) and (SE2) respectively
11

4)Calculate the standard error of the difference
between the 2 sample estimates as
√[(SE1)2
+ (SE2)2
]
5)Calculate the critical ratio (CR) or z value.
z= difference between sample estimates/ SE of
difference
12
Steps in tests for large sample(contd..)

6)Refer to the normal distribution table &
corresponding to this calculated value of z, find the
value of probability P.
7)If P is ≤ 0.05, reject the null hypothesis and
conclude that the difference between the 2 sample
estimates as significant.
13
Steps in tests for large sample(contd..)

Tests for large samples
SE of mean= s/ √n
SE of proportion= √(pq/n)
SE of difference between 2 means= √[(s1
2
/n1 ) +
(s2
2
/n2)]
SE of difference between 2 proportions= √[(p1q1/n1)
+ (p2q2/n2)]
14

Example
In an epidemic of gastroenteritis in an area the number of cases reported in 2
populations consuming water from different sources were as follows:
Source of
water
No. of people consuming
water from a particular source
No. of cases of
gastroenteritis
Tap water 800 35
Hand pump 2400 120
Total 3200 155
We have to find out whether the difference in the proportion of cases in the 2
groups is significantly different or not.
15

Tests for small samples
Assumptions for t- distribution:
the means of the 2 samples are normally distributed
the means of the 2 samples are independently
distributed
 the variances of the 2 samples are equal.
16

t- test for paired observation
The formula to calculate t statistics is,
t= d¯/smd with (n-1) degrees of freedom
n= number of observations in the sample
d¯= mean of differences in the values of the
variable of the sample observations before & after
treatment
17

smd= SE of mean difference
smd= sd/ √n
 sd= standard deviation of the values of d (the
differences in variable before & after the treatment)
18

1)Set up null hypothesis that d¯=0
2)Calculate the difference d1 for each pair of observations
before & after treatment and compute their mean d¯
3)Calculate the SD of these differences
sd= [√{sum total of difference between
individual observation & d¯}2
]/(n-1)
n= number of pairs of observations
4)Calculate SE smd from the formula
smd= sd/ √n
19

5)Calculate the value of t- statistics as t= d¯/smd
6)Compute the degrees of freedom as (n-1)
7)From the t- distribution table find the probability
level corresponding to this value of t & with degrees
of freedom (n-1)
20

Patient no.
Hb level in gms%
Difference
between
after & before
therapy (d)
(d-d¯) (d-d¯)2
Before
therapy
After therapy
1 5.6 10.2 4.6 1.87 3.50
2 4.8 9.4 4.6 1.87 3.50
3 6.5 11.0 4.5 1.77 3.13
4 7.5 7.5 0.0 -2.73 7.45
5 4.5 7.5 3.0 -0.27 0.07
6 3.5 6.0 2.5 -0.23 0.05
7 6.7 8.0 1.3 -1.43 2.04
8 6.2 9.6 3.4 0.67 0.45
9 5.6 10.0 4.4 1.67 2.79
10 4.4 8.4 4.0 1.27 1.61
11 7.5 8.0 0.5 -2.23 4.97
12 8.0 8.0 0.0 -2.73 7.45
TOTAL 32.8 37.01
21

Unpaired t-test
t= {(x1¯- x2¯)/ smd}
smd is estimated standard error of the difference between the 2
sample means
smd= √[{(n1+n2)/n1n2}{(n1-1)s1
2
+ (n2-1)s2
2
}/ (n1+n2-2)]
s1
2
and s2
2
are the SD of 2 samples
n1 and n2 are respective sample sizes
t= (x1¯- x2¯)/ √[{(n1+n2)/n1n2}{(n1-1)s1
2
+ (n2-1)s2
2
}/ (n1+n2-2)]
with (n1 + n2 – 2) degrees of freedom
22

Inanexperimenttoknowwhetherthereisanydifferenceinthelengthofsmall
intestinesbetween malesandfemales,theobservationsrecordedwere asfollows:
Males Females
No.ofobservations 17 15
Meanlengthofthesmallintestines 157 146
SDoftheobservations 34 31
23

ANOVA
Assumptions:
the effects under different groups are additive
Sample of observations has been drawn using
random sampling procedure
Samples come from normally distributed population
the variance is same in various groups
24

One way ANOVA
1)Calculate sum of observations of each group (village)=
(Ti)
2)Calculate sum of all observations (Σxij),where xij
represents each observation
3)Calculate square of sum of all observations= (Σxij)2
4)Calculate total sum of squares, ie = [Σ(xij
2
)- {(Σxij)2
/n}];
n= total no. of observations.
(Σxij)2
/n is called correction factor(CF)
25

5)Calculate ‘sum of squares between groups’, ie
= {Σ(Tj
2
/ki)-CF}; Tj= sum of observations in each
group; ki= no. of observations in each group
6)Sum of squares within groups is obtained as
difference between the ‘total sum of squares’ and ‘sum
of squares between groups’ ie [(Σxij)2 – CF] – [Σ(Tj
2
/ki)
– CF] = (Σxij)2
- Σ(Tj
2
/ki).
26
One way ANOVA

ANALYSIS OF VARIANCE TABLE
Source of
sum of squares
Degree of
freedom
Sum of
squares
Mean sum
of squares
F
Between villages 3 349.525 116.5083 4.544
Within villages 36 922.950 25.6375
Total 39 1272.475
27

Two way ANOVA
 To study the differences within sub classification of the
groups also.
28

Analysis of variance table
Source of
SS
df Sum of squares MSS F
Between villages 9 134.17 14.908 0.855
Between
trimesters
2 436.72 218.36 12.526
Residual 18 313.78 17.432
Total 29 884.67
29

NON-PARAMETRIC TESTS- Assumptions
 The observations do not follow normal distribution
 The populations do not have same variances
 The observations may or may not be independent
 Population parameters involved are median, mode
 Data may be nominally or ordinally scaled
Example: nominal(male/female); ordinal(good-better-
best)
30

NON-PARAMETRIC TESTS…
 Chi-square test
 Fisher’s exact test
 The Sign test
 Wilcoxon’s Signed Rank test
 Mann Whitney U test
 Kruskal- Wallis H test
 Friedman test
 Mc Nemar test
31

Chi-square test
χ2
= Σ {(observed frequency – expected frequency)2
/
expected frequency}
This value is calculated for each cell in the table & sum
of above ratios is the total chi-square value.
 df = (c-1)(r-1)
32

Blood
group
Non
leprosy
Lepromatous
leprosy
Non
Lepromatous
Leprosy
A
(131/471)x150
= 41.7
(131/471)x169
= 47.0
(131/471)x152
= 42.3
B
(145/471)x150
= 46.2
(145/471)x169
= 52.0
(145/471)x152
= 46.8
O
(154/471)x150
= 49.0
(154/471)x169
= 55.3
(154/471)x152
= 49.7
AB
(41/471)x150
= 13.1
(41/471)x169
= 14.7
(41/471)x152
= 13.2
33

Blood group
Non
leprosy
Lepromatous
leprosy
Non lepromatous
leprosy
Total
A 3.28 0.09 2.22 5.59
B 4.12 0.17 2.49 6.78
O 0.08 0.25 0.06 0.39
AB 0.00 0.50 0.59 1.09
Total 7.48 1.01 5.36 13.85
34

Chi-square test for a 2x2 table
 χ2
= [(ad-bc)2
x G] / [(a+b)(c+d)(a+c)(b+d)]
df = (2-1)(2-1) = 1
35

Filariasis
infestation
Male Female Total
Yes 28 (a) 20(b) 48
No 237(c) 222(d) 459
Total 265 242 507(G)
36

Fisher’s Exact probability test
P = [{(a+c)! (b+d)! (a+b)! (c+d)! }/ {n! a! b! c! d!}]
a, b, c & d = cell frequencies in 2x2 table
 n = total number of observations
37

Habit Boys Girls Total
Exercise
regularly
2 8 10
Do not exercise
regularly
10 4 14
Totals 12 12 24
38

Sign test
The significance of difference can be tested using
usual χ2
test, which is as follows:
χ2
= [(|a-b| - 1)2
] / n with 1 df
a & b = no. of (+) & (-) respectively
n = (a+b)
39

All ‘0’ differences, ie when the 2 tests are identical in
their results are omitted for calculation so that ‘n’ is
always equal to (a+b).
P is obtained from χ2
distribution table & the conclusion
about the significance is made as usual
40
Sign test

Patient no. Serology Parasitology Difference
1 1 0 +
2 1 0 +
3 0 1 -
4 1 0 +
5 1 0 +
6 0 0 0
7 1 0 +
8 1 0 +
9 0 0 0
10 0 1 -
11 0 1 -
12 1 1 0
41

Wilcoxon’s Sign Rank test
This is useful in testing the significance of
differences in paired observations, when the data is
quantitative in nature.
42

Serum FDP values in µgm/ml
Case no.
Before
operation
After
operation
Difference
Signed
rank
1 5.0 7.8 -2.8 -2
2 10.0 180.0 -170.0 -11
3 18.0 10.0 +8.0 +5
4 5.0 80.0 -75.0 -10
5 10.0 15.0 -5.0 -3.5
6 20.0 10.0 +10.0 +6
7 5.0 180.0 -175.0 -12
8 2.5 40.0 -37.5 -8
9 15.0 10.0 +5.0 +3.5
10 10.0 7.5 +2.5 +1
11 80.0 10.0 +70.0 +9
12 5.0 20.0 -15.0 -7
43

Mann Whitney U test
This is useful in testing differences between
unpaired observations.
44

FDP values in µg/ ml
Control
group
10.0 180.0 80.0 5.0 40.0 15.0 30.0 160.0
Treated
group
7.8 40.0 10.0 80.0 180.0 5.0 80.0 10.0
5.0 7.5
45

FDP
Values
5.0 5.0 5.0 7.5 7.8 10.0 10.0 10.0 15.0 30.0
Ranks 2 2 2 4 5 7 7 7 9 10
FDP
values
40.0 40.0 80.0 80.0 80.0 160.0 180.0 180.0
Ranks 11.5 11.5 14 14 14 16 17.5 17.5
46

Kruskal Wallis H test
This is the non-parametric equivalent to F test used in
one way ANOVA. This test is used to test whether or not
the groups of independent samples have been drawn
from the same population. It is calculated using formula:
H = [12/n(n+1)]{Σ(Ri
2
/ni)} – 3(n+1)
47

Group 1 Group 2 Group 3
Score Rank Score Rank Score Rank
10 1 11 2 30 20
12 3 14 4 32 22
15 5 22 12 24 14
17 7 28 18 31 21
23 13 20 10 26 16
18 8 16 6 34 24
19 9 29 19 27 17
21 11 33 23
25 15
n1 = 7 R1 = 46 n2 = 8 R2 = 82 n3 = 9 R3 = 172
48

Friedman Test
This is the non-parametric equivalent of two way
ANOVA, where each group may have further sub-
divisions. df = k-1
The formula used is as follows:
χ2
= [{(12) x Σ(Rj
2
)} / {nk(k+1)}] – {3n(k+1)}
ΣRj = sum of ranks in each column
n = no. of rows
k = no. of columns
49

Group
of
care
givers
Need 1 Need 2 Need 3 Need 4
Score Rank Score Rank Score Rank Score Rank
Group
I
13 2 9 4 10 3 16 1
Group
II
10 2 4 3 3 4 12 1
Group
III
12 1 6 3 2 4 10 2
Rank
sums
5 10 11 4
50

Mc Nemar test
 It is a type of 2x2 chi-square test. It is for comparisons
of variables from matched pairs & uses information
only from discordant pairs(variables are not
independent).
 Reduces type 1 error
χ2
= [(|f-g| - 1)2
] / (f+g)
 df = 1
51

Alcohol addiction
(cases)
Total
Yes No
Alcohol
addiction
(controls)
Yes 10 30 40
No 20 40 60
Total 30 70 100
52

Non-parametric tests: Advantages over
parametric tests
 Can be used to test the data measured on nominal
& ordinal scales
 Easier to compute, understand & explain
 Make fewer assumptions about distribution of
samples
 Can be used even for very small sample sizes
 The computational burden is so light that they are
known as “short cut” methods
53

Advantages…
 Need not involve population parameters ie can be
used to analyze the data from populations where
mean & standard deviations are not available or
undeterminable
 Results may be as exact as parametric procedures
when used even on populations following normal
distribution
54

Non-parametric tests: Disadvantages
 Can test the statistical hypothesis but cannot
estimate the parameter
 Low power of tests
 Difficult to compute by hand for large samples
 Tables are not widely available
55

Disadvantages…
 Waste of time & data if all assumptions of a
statistical model are satisfied & if there is a
suitable parametric test
56

NON-PARAMETRIC COMPLEMENTS OF
PARAMETRIC TESTS
PARAMETRIC TESTS NON-PARAMETRIC
TESTS
Paired t test Wilcoxon’s signed rank
test
Unpaired t test Mann Whitney U test
One way ANOVA Kruskal Wallis test
Two way ANOVA Friedman test
57

10/16/2024
Clinical significance v/s statistical significance
58
 A possible antipyretic is tested in patients with fever
500 received the drug
500 received the placebo
 Temperatures measured 4 hours after dosing
 P value= 0.011
 Statistical significance ?Yes
 Clinical significance? No .Temperature only fell by about 0.1 degree
Celsius
N Mean St Dev SE
Mean
Drug 500 39.950 0.653 0.029
Control 500 40.058 0.699 0.031

10/16/2024
Conclusion
62
 Tests of significance draws the conclusions about
population based on the results observed in the
random sample.
 So, we should be careful in choosing the method
which suits the hypothesis bearing in mind the
assumptions of the same.

10/16/2024
References
63
1] Rao NSN, Murthy NS. Applied statistics in health
sciences. 1st
ed. New Delhi: Jaypee Brothers Medical
Publishers (P) Ltd; 2008.
2] Dixit JV. Principles and practice of biostatistics. 3rd
ed.
Jabalpur: M/S Banarsidas Bhanot Publishers; 2005.
3] Hennekens CH, Buring JE. Epidemiology in medicine.
1st
ed. Boston, Toronto: Little Brown & Company.
4] Sundaran KR, Dwivedi SN,Sreenivas V. Medical
Statistics-Principles and methods. New Delhi: BI
Publications Pvt Ltd.;2010

10/16/2024
64
5] Kapur JN, Saxena HC. Mathematical Statistics.4th
Edition. Delhi: Chand and Co Publishers:1967
6]Jeyaseelan L. Tests of Significance notes.
Fundamentals in Biostatistics and SPSS. 2014
7]Dr Farah Naaz Fathima. Seminar notes on Tests of
significance. Presented on 2/11/2004
8] Dr Akshay K M. Seminar notes on Parametric tests.
Presented on 26/11/2008
9] Dr Bhanu M . Seminar notes on Tests of Significance.
Presented on 29/04/2011.
10] Aleyamma Mathew. ABC of Medical Statistics.
Kerala Surgical journal, 1999:6(1);p 1-71

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