This document discusses various statistical tests used to analyze dental research data, including parametric and non-parametric tests. It provides information on tests of significance such as the t-test, Z-test, analysis of variance (ANOVA), and non-parametric equivalents. Key points covered include the differences between parametric and non-parametric tests, assumptions and applications of the t-test, Z-test, ANOVA, and non-parametric alternatives like the Mann-Whitney U test and Kruskal-Wallis test. Examples are provided to illustrate how to perform and interpret common statistical analyses used in dental research.
• Non parametric tests are distribution free methods, which do not rely on assumptions that the data are drawn from a given probability distribution. As such it is the opposite of parametric statistics
• In non- parametric tests we do not assume that a particular distribution is applicable or that a certain value is attached to a parameter of the population.
When to use non parametric test???
1) Sample distribution is unknown.
2) When the population distribution is abnormal
Non-parametric tests focus on order or ranking
1) Data is changed from scores to ranks or signs
2) A parametric test focuses on the mean difference, and equivalent non-parametric test focuses on the difference between medians.
1) Chi – square test
• First formulated by Helmert and then it was developed by Karl Pearson
• It is both parametric and non-parametric test but more of non - parametric test.
• The test involves calculation of a quantity called Chi square.
• Follows specific distribution known as Chi square distribution
• It is used to test the significance of difference between 2 proportions and can be used when there are more than 2 groups to be compared.
Applications
1) Test of proportion
2) Test of association
3) Test of goodness of fit
Criteria for applying Chi- square test
• Groups: More than 2 independent
• Data: Qualitative
• Sample size: Small or Large, random sample
• Distribution: Non-Normal (Distribution free)
• Lowest expected frequency in any cell should be greater than 5
• No group should contain less than 10 items
Example: If there are two groups, one of which has received oral hygiene instructions and the other has not received any instructions and if it is desired to test if the occurrence of new cavities is associated with the instructions.
2) Fischer Exact Test
• Used when one or more of the expected counts in a 2×2 table is small.
• Used to calculate the exact probability of finding the observed numbers by using the fischer exact probability test.
3) Mc Nemar Test
• Used to compare before and after findings in the same individual or to compare findings in a matched analysis (for dichotomous variables).
Example: comparing the attitudes of medical students toward confidence in statistics analysis before and after the intensive statistics course.
4) Sign Test
• Sign test is used to find out the statistical significance of differences in matched pair comparisons.
• Its based on + or – signs of observations in a sample and not on their numerical magnitudes.
• For each subject, subtract the 2nd score from the 1st, and write down the sign of the difference.
It can be used
a. in place of a one-sample t-test
b. in place of a paired t-test or
c. for ordered categorial data where a numerical scale is inappropriate but where it is possible to rank the observations.
5) Wilcoxon signed rank test
• Analogous to paired ‘t’ test
6) Mann Whitney Test
• similar to the student’s t test
7) Spearman’s rank correlation - similar to pearson's correlation.
OBJECTIVES:
Run the test of hypothesis for mean difference using paired samples. Construct a confidence interval for the difference in population means using paired samples.
Observation of interest will be the difference in the readings
before and after intervention called paired difference observation.
Paired t test:
A paired t-test is used to compare two means where you have two samples in which observations in one sample can be paired with observations in the other sample.
Examples of where this might occur are:
Before-and-after observations on the same subjects (e.g. students’ test
results before and after a particular module or course).
A comparison of two different methods of measurement or two different treatments where the measurements/treatments are applied to the same subjects (e.g. blood pressure measurements using a sphygmomanometer and a dynamap).
When there is a relationship between the groups, such as identical twins.
This test is concerned with the pair-wise differences
between sets of data.
This means that each data point in one group has a related data point in the other group (groups always have equal numbers).
ASSUMPTIONS:
The sample or samples are randomly selected
The sample data are dependent
The distribution of differences is approximately normally
distributed.
Note: The under root is onto the entire numerator and denominator, so you should take the root after solving it entirely
where “t” has (n-1) degrees of freedom and “n” is
the total number of pairs.
• Non parametric tests are distribution free methods, which do not rely on assumptions that the data are drawn from a given probability distribution. As such it is the opposite of parametric statistics
• In non- parametric tests we do not assume that a particular distribution is applicable or that a certain value is attached to a parameter of the population.
When to use non parametric test???
1) Sample distribution is unknown.
2) When the population distribution is abnormal
Non-parametric tests focus on order or ranking
1) Data is changed from scores to ranks or signs
2) A parametric test focuses on the mean difference, and equivalent non-parametric test focuses on the difference between medians.
1) Chi – square test
• First formulated by Helmert and then it was developed by Karl Pearson
• It is both parametric and non-parametric test but more of non - parametric test.
• The test involves calculation of a quantity called Chi square.
• Follows specific distribution known as Chi square distribution
• It is used to test the significance of difference between 2 proportions and can be used when there are more than 2 groups to be compared.
Applications
1) Test of proportion
2) Test of association
3) Test of goodness of fit
Criteria for applying Chi- square test
• Groups: More than 2 independent
• Data: Qualitative
• Sample size: Small or Large, random sample
• Distribution: Non-Normal (Distribution free)
• Lowest expected frequency in any cell should be greater than 5
• No group should contain less than 10 items
Example: If there are two groups, one of which has received oral hygiene instructions and the other has not received any instructions and if it is desired to test if the occurrence of new cavities is associated with the instructions.
2) Fischer Exact Test
• Used when one or more of the expected counts in a 2×2 table is small.
• Used to calculate the exact probability of finding the observed numbers by using the fischer exact probability test.
3) Mc Nemar Test
• Used to compare before and after findings in the same individual or to compare findings in a matched analysis (for dichotomous variables).
Example: comparing the attitudes of medical students toward confidence in statistics analysis before and after the intensive statistics course.
4) Sign Test
• Sign test is used to find out the statistical significance of differences in matched pair comparisons.
• Its based on + or – signs of observations in a sample and not on their numerical magnitudes.
• For each subject, subtract the 2nd score from the 1st, and write down the sign of the difference.
It can be used
a. in place of a one-sample t-test
b. in place of a paired t-test or
c. for ordered categorial data where a numerical scale is inappropriate but where it is possible to rank the observations.
5) Wilcoxon signed rank test
• Analogous to paired ‘t’ test
6) Mann Whitney Test
• similar to the student’s t test
7) Spearman’s rank correlation - similar to pearson's correlation.
OBJECTIVES:
Run the test of hypothesis for mean difference using paired samples. Construct a confidence interval for the difference in population means using paired samples.
Observation of interest will be the difference in the readings
before and after intervention called paired difference observation.
Paired t test:
A paired t-test is used to compare two means where you have two samples in which observations in one sample can be paired with observations in the other sample.
Examples of where this might occur are:
Before-and-after observations on the same subjects (e.g. students’ test
results before and after a particular module or course).
A comparison of two different methods of measurement or two different treatments where the measurements/treatments are applied to the same subjects (e.g. blood pressure measurements using a sphygmomanometer and a dynamap).
When there is a relationship between the groups, such as identical twins.
This test is concerned with the pair-wise differences
between sets of data.
This means that each data point in one group has a related data point in the other group (groups always have equal numbers).
ASSUMPTIONS:
The sample or samples are randomly selected
The sample data are dependent
The distribution of differences is approximately normally
distributed.
Note: The under root is onto the entire numerator and denominator, so you should take the root after solving it entirely
where “t” has (n-1) degrees of freedom and “n” is
the total number of pairs.
linearity concept of significance, standard deviation, chi square test, stude...KavyasriPuttamreddy
Linearity concept of significance, standard deviation, chi square test, students T- test, ANOVA test , pharmaceutical science, statistical analysis, statistical methods, optimization technique, modern pharmaceutics, pharmaceutics, mpharm 1 unit i sem, 1 year m
pharm, applications of chi square test, application of standard deviation , pharmacy, method to compare dissolution profile, statistical analysis of dissolution profile, important statical analysis, m. pharmacy, graphical representation of standard deviation, graph of chi square test, graph of T test , graph of ANOVA test ,formulation of t test, formulation of chi square test, formula of standard deviation.
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SlideShare Description for "Chatty Kathy - UNC Bootcamp Final Project Presentation"
Title: Chatty Kathy: Enhancing Physical Activity Among Older Adults
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2. Contents
• Introduction
• Tests of significance
• Parametric test
• Non Parametric test
• Application of tests in dental research
• Conclusion
• Bibliography
3. • VARIABLE
– A general term for any feature of the unit
which is observed or measured.
• FREQUENCY DISTRIBUTION
– Distribution showing the number of
observations or frequencies at different values or
within certain range of values of the variable.
4. Tests of significance
• Population
• is any finite collection of elements
I.e – individuals, items, observations etc,.
✓Statistic –
✓is a quantity describing a sample, namely a function
of observations
✓Parameter –
✓ is a constant describing a population
✓Sample –
✓is a part or subset of the population
6. Hypothesis testing
• Hypothesis
• is an assumption about the status of a phenomenon
0
H
H
✓Null hypothesis or hypothesis of no difference –
✓States no difference between statistic of a sample & parameter
of population or b/n statistics of two samples
✓This nullifies the claim that the experiment result is different
from or better than the one observed already
✓Denoted by
7. • Alternate hypothesis –
• Any hypothesis alternate to null hypothesis, which is to be
tested
• Denoted by
1
H
Note : the alternate hypothesis is accepted when
null hypothesis is rejected
8. Type I & type II errors
• Type I error =
• Type II error =
No error
Type II error
is true
Type I error
No error
is true
Accept
Accept
0
H 1
H
1
H
0
H
When primary concern of the test is to see
whether the null hypothesis can be rejected
such test is called Test of significance
9. α ERROR
• The probability of committing type I error is called “P”
value
• Thus p-value is the chance that the presence of difference
is concluded when actually there is none
✓Type I error – important- fixed in advance at a
low level
✓Thus α is the maximum tolerable probability
of type I error
10. • Difference b/n level of significance & P-value -
LOS P-value
1) Maximum tolerable
chance of type I error
1) Actual probability of
type I error
2) α is fixed in advance 2) calculated on basis of data
following procedures
The P-value can be more than α or less than α depending on data
When P-value is < than α → results is statistically significant
11. • The level of significance is usually fixed at 5%
(0.05) or 1% (0.01) or 0.1% (0.001) or 0.5% (0.005)
• Maximum desirable is 5% level
• When P-value is b/n
0.05-0.01 = statistically significant
< than 0.01= highly statistically significant
Lower than 0.001 or 0.005 = very highly significant
12. Sampling Distribution
1/2
1/2
Zone of
Rejection H0
Zone of
Rejection H0
Zone of
Acceptance H0
SD
x 96
.
1
+
SD
x 96
.
1
−
x
Confidence limits – 95%
Confidence interval
13. TESTS IN TEST OF SIGNIFICANCE
Parametric
(normal distribution &
Normal curve )
Non-parametric
(not follow
normal distribution)
Quantitative data Qualitative data
1) Student ‘t’test
1) Paired
2) Unpaired
2) Z test
(for large samples)
3) One way ANOVA
4) Two way ANOVA
1) Z – prop test
2) χ² test
Qualitative
(quantitative converted
to qualitative )
1. Mann Whitney U test
2. Wilcoxon rank test
3. Kruskal wallis test
4. Friedmann test
14. Dr Sandesh N
Parametric Uses Non-parametric
Paired t test Test of diff b/n
Paired observation
Wilcoxon signed
rank test
Two sample t test Comparison of two
groups
Wilcoxon rank sum test
Mann Whitney U test
Kendall’s s test
One way Anova Comparison of
several groups
Kruskal wallis test
Two way Anova Comparison of groups
values on two variables
Friedmann test
Correlation
coefficient
Measure of association
B/n two variable
Spearman’s rank
Correlation
Kendall’s rank
correlation
Normal test (Z test ) Chi square test
15. Student ‘t’ test
• Small samples do not follow normal distribution
• Prof W.S.Gossett – Student‘t’ test – pen name – student
• It is the ratio of observed difference b/n two mean of small samples
to the SE of difference in the same
17. • Criteria for applying ‘t’ test –
• Random samples
• Quantitative data
• Variable follow normal distribution
• Sample size less than 30
• Application of ‘t’ test –
1. Two means of small independent sample
2. Sample mean and population mean
3. Two proportions of small independent samples
18. Unpaired ‘t’ test
I) Difference b/n means of two independent samples
Group 1 Group 2
Sample size
Mean
SD
1
n 2
n
1
x 2
x
1
SD 2
SD
( ) 0
2
1
0 =
−
x
x
H
( ) 0
2
1
1
−
x
x
H
1) Null hypothesis
2) Alternate hypothesis
Data –
19. 3) Test criterion
( )
2
1
2
1
x
x
SE
x
x
t
−
−
=
( ) by
calculated
is
of
here 2
1 x
x
SE −
( )
+
=
−
2
1
2
1
1
1
of
n
n
SD
x
x
SE
( ) ( )
2
1
1
where
2
1
2
2
2
2
1
1
−
+
−
+
−
=
n
n
SD
n
SD
n
SD
( ) ( ) ( )
+
−
+
−
+
−
=
−
2
1
2
1
2
2
2
2
1
1
2
1
1
1
2
1
1
n
n
n
n
SD
n
SD
n
x
x
SE
20. 4) Calculate degree of freedom
( ) ( ) 1
1
1 2
1
2
1 −
+
=
−
+
−
= n
n
n
n
df
6) Draw conclusions
5) Compare the calculated value &
the table value
21. • Example – difference b/n caries experience of high
& low socioeconomic group
Sl
no
Details High socio
economic group
Low socio
economic group
I Sample size
II DMFT
III Standard deviation
15
1 =
n 10
2 =
n
91
.
2
1 =
x 26
.
2
2 =
x
27
.
0
1 =
SD 22
.
0
2 =
SD
( )
23
,
34
.
6
1027
.
0
65
.
0
2
1
2
1
=
=
=
−
−
= df
x
x
SE
x
x
t
001
.
0
001
.
0 76
.
3 t
t
t c
=
There is a significant difference
22. Other applications
II) Difference b/n sample mean & population mean
n
SD
SE
x
t
=
−
=
1
−
= n
df
+
−
=
2
1
2
1
1
1
n
n
PQ
p
p
t
2
1
2
2
1
1
where
n
n
p
n
p
n
P
+
+
=
P
Q −
=1
2
2
1 −
+
= n
n
df
III) Difference b/n two sample proportions
23. Paired ‘t’ test
• Is applied to paired data of observations from one
sample only when each individual gives a paired of
observations
• Here the pair of observations are correlated and not
independent, so for application of ‘t’ test following
procedure is used-
1. Find the difference for each pair
2. Calculate the mean of the difference (x) ie
3. Calculate the SD of the differences & later SE
x
y
y =
− 2
1
x
=
n
SD
SE
24. 4. Test criterion
( ) ( )
n
x
SD
x
d
SE
x
t =
−
=
0
1
−
= n
df
7. Draw conclusions
6. Refer ‘t’ table & find the probability
of calculated value
5. Degree of freedom
25. • Example – to find out if there is any significant improvement in DAI
scores before and after orthodontic treatment
Sl no DAI before DAI after Difference Squares
1 30 24 6 36
2 26 23 3 9
3 27 24 3 9
4 35 25 10 100
5 25 23 2 4
Total 20 158
26. ( ) ( ) ( ) ( ) ( ) ( )2
2
2
2
2
2
4
2
4
10
4
3
4
3
4
6
squares,
of
sum
4
5
20
−
+
−
+
−
+
−
+
−
=
−
=
=
=
x
x
n
x
x
Mean
46
4
36
1
1
4
=
+
+
+
+
=
( )
78
.
2
but
4
1
6352
.
2
5179
.
1
4
5179
.
1
5
391
.
3
391
.
3
5
.
11
4
46
1
5
.
0
5
.
0
2
t
t
t
n
df
SE
x
t
n
SD
SE
n
x
x
SD
c
c
=
=
−
=
=
=
=
=
=
=
=
=
=
−
−
=
Hence not significant
27. Z test (Normal test)
• Similar to ‘t’ test in all aspect except that the sample size should be >
30
• In case of normal distribution, the tabulated value of Z at -
960
.
1
level
%
5 05
.
0 =
= Z
576
.
2
level
%
1 01
.
0 =
= Z
290
.
3
level
%
1
.
0 001
.
0 =
= Z
28. • Z test can be used for –
1. Comparison of means of two samples –
+
=
2
2
2
1
2
1
n
SD
n
SD
( )
2
1
2
1
x
x
SE
x
x
Z
−
−
=
( ) ( )
2
2
2
1
2
1
where SE
SE
x
x
SE +
=
−
n
SD
x
Z
2
−
=
2. Comparison of sample mean & population mean
29. 3. Difference b/n two sample proportions
2
1
2
2
1
1
2
1
2
1
here
w
1
1 n
n
p
n
p
n
P
n
n
PQ
p
p
Z
+
+
=
+
−
=
P
Q −
=1
−
=
n
PQ
P
p
Z
1
Where p = sample proportion
P = populn proportion
4. Comparison of sample proportion
(or percentage) with population proportion
(or percentage)
30. Analysis of variance (ANOVA)
• Useful for comparison of means of several groups
• R A Fisher in 1920’s
• Has four models
1. One way classification (one way ANOVA )
2. Single factor repeated measures design
3. Nested or hierarchical design
4. Two way classification (two way ANOVA)
31. One way ANOVA
• Can be used to compare like-
• Effect of different treatment modalities
• Effect of different obturation techniques on the apical seal , etc,.
32. Groups (or treatments) 1 2 i k
Individual values
Calculate
No of observations
Sum of x values
Sum of squares
Mean of values
11
x
2
i
x
22
x
n
x2
n
x1
12
x
1
k
x
1
i
x
21
x
in
x
2
k
x
kn
x
n n n n
n
x
x
x 1
12
11 ...+
+
+
=
1
Τ 2
T i
T k
T
( ) ( ) ( )2
1
2
12
2
11 .. n
x
x
x +
+
+
=
1
S
2
S i
S k
S
n
T
x 1
1 = 2
x i
x k
x
33. ANOVA table
Sl
no
Source
of
variation
Degree
of
freedom
Sum of squares Mean sum of
squares
F ratio or
variance ratio
I Between
Groups
II With in
groups
III Total
1
−
k
k
n−
1
−
n
( )
−
=
−
i i
i i
N
T
x
x
x
2
2
2
( )
−
=
− i
i
i
i j ij
i j i
ij
n
T
x
x
x
2
2
2
( )
−
=
−
N
T
x
x
x i j ij
i j ij
2
2
2
( )
1
2
2
−
−
=
k
x
x
S i i
B
k
N
n
T
x
S
i j i
i
i
ij
W
−
−
=
2
2
2
1
2
2
2
−
−
=
N
N
T
x
S
i j ij
T
( )
k
N
k
S
S
W
B
−
− ,
1
2
2
34. Example- see whether there is a difference in number of patients
seen in a given period by practitioners in three group practice
Practice A B C
Individual values 268 387 161
349 264 346
328 423 324
209 254 293
292 239
Calculate
No of observations (n) 5 4 5
Sum of x values 1441 1328 1363
Sum of squares 426899 462910 393583
Mean of values 288.2 332.0 272.6
36. Two way ANOVA
• Is used to study the impact of two factors on
variations in a specific variable
• Eg – Effect of age and sex on DMFT value
37. Sample values
blocks Treatments sample size Total Mean
value
i
ii
..
n
Sample
size
Total
Mean
value
11
x
32
x
22
x
n
x2
n
x1
12
x
1
k
x
31
x
21
x
n
x3
2
k
x
kn
x
n n n n
k
k
k
N
nk =
2
T
n
T
1
T
1
T
2
T
k
T
3
T T
2
x
1
x
k
x
3
x
2
x
1
x
n
x
x
38. Non parametric tests
• Here the distribution do not require any specific pattern of
distribution. They are applicable to almost all kinds of distribution
• Chi square test
• Mann Whitney U test
• Wilcoxon signed rank test
• Wilcoxon rank sum test
• Kendall’s S test
• Kruskal wallis test
• Spearman’s rank correlation
39. Chi square test
• By Karl Pearson & denoted as χ²
• Application
1. Alternate test to find the significance of difference in two or more than two
proportions
2. As a test of association b/n two events in binomial or multinomial samples
3. As a test of goodness of fit
40. • Requirement to apply chi square test
• Random samples
• Qualitative data
• Lowest observed frequency not less than 5
• Contingency table
• Frequency table where sample classified according to two
different attributes
• 2 rows ; 2 columns => 2 X 2 contingency table
• r rows : c columns => rXc contingency table
( )
−
=
E
E
O
2
2
O – observed frequency
E – expected frequency
41. • Steps
1. State null & alternate hypothesis
2. Make contingency table of the data
3. Determine expected frequency by
4. Calculate chi-square of each by-
( )
c
r
( )
frequency
total
N
c
r
E
=
( )
E
E
O
2
2 −
=
42. 5. calculate degree of freedom
6. Sum all the chi-square of each cell – this gives
chi-square value of the data
7. Compare the calculated value with the table
value at any LOS
8. Draw conclusions
( )
−
=
E
E
O
2
2
( )( )
1
1 −
−
= r
c
df
43. • Chi square test only tells the presence or absence of
association
• but does not measure the strength of association
44. Wilcoxon signed rank test
• Is equivalent to paired ‘t’ test
• Steps
• Exclude any differences which are zero
• Put the remaining differences in ascending order, ignoring the signs
• Gives ranks from lowest to highest
• If any differences are equal, then average their ranks
• Count all the ranks of positive differences – T+
• Count all the ranks of negative differences – T-
45. • If there is no differences b/n variables then T+ & T_ will
be similar, but if there is difference then one sum will
be large and the other will be much smaller
• T= smaller of T+&T_
• Compare the T value with the critical value for 5%, 2%
& 1% significance level
• A result is significant if it is smaller than critical value
46. Mann Whitney U test
• Is used to determine whether two independent sample have been
drawn from same sample
• It is a alternative to student ‘t’ test & requires at least ordinal or
normal measurement
( )
2
1
1
1
2
1
2
1
R
or
R
n
n
n
n
U −
+
+
=
Where, n1n2 are sample sizes
R1 R2 are sum of ranks assigned to I & II group
47. Comparison of birth weights of children born to 15 non
smokers with those of children born to 14 heavy smokers
NS 3.9 3.7 3.6 3.7 3.2 4.2 4.0 3.6 3.8 3.3 4.1 3.2 3.5 3.5 2.7
HS 3.1 2.8 2.9 3.2 3.8 3.5 3.2 2.7 3.6 3.7 3.6 2.3 2.3 3.6
R1 26 23 16 21 8 29 27 17 24 12 28 10 15 13 03
R2 7 5 6 11 25 14 9 4 20 22 19 2 1 18
Ranks assignments
48. Sum of R1= 272 and Sum of R2=163
Difference T=R1 – R2 is 109
The table value of T0.05 is 96 , so reject the H0
We conclude that weights of children born to the
heavy smokers are significantly lower than those of
the children born to the non-smokers (p<0.05)
50. Research interested in relationship
B/n more than two variables
Use multiple regression
Or
Multivariate analysis
Multiple – variable problem
51. Bibliography
• Biostatistics
– Rao K Vishweswara, Ist edition.
• Methods in Biostatistics
– Dr Mahajan B K, 5th edition.
• Essentials of Medical Statistics
– Kirkwood Betty R, 1st edition.
• Health Research design and Methodology
– Okolo Eucharia Nnadi.
• Simple Biostaistics
– Indrayan,1st edition.
• Statistics in Dentistry
– Bulman J S