SlideShare una empresa de Scribd logo
1 de 70
Descargar para leer sin conexión
TESTS OF SIGNIFICANCE
Moderator: Dr. S.K.Bhasin
Presenters: Migom
Parnava
Kartikey
1
Probability distribution
1. Normal distribution
2. Binomial distribution
3. Poissons distribution
2
Normal curve
• Gaussian Distribution
• Continuous
• Bell shaped
• Symmetrical
• Mean, Mode and Median coincide
3
Confidence level and Confidence limits
4
Z distribution
• Z transformation: Z= observation – mean = X - 
SD 
5
• P value:
• The probability of observing a result as extreme as or more extreme than the one
actually observed from chance alone (i.e., if the null hypothesis is true)
• Power of a test:
• Probability that a study or a trial will be able to detect a specified difference
• Power= 1- β
6
Scales of measurement
• Nominal scales
• E.g. outcome of surgical procedure
• Ordinal scales
• E.g. APGAR score, tumour staging
• Numerical scale
• E.g. Age, weight, number of fractures
7
Measures of central tendency
• Mean
• Median
• Mode
Some important concepts:
• Standard deviation
• Variance
• Standard error of mean
8
TESTS OF STATISTICAL SIGNIFICANCE
• Procedure for comparing observed data with a claim (hypothesis) whose
truth we want to assess
• Appropriate test of significance:
• Data
• Sample
• Purpose
9
Parametric tests Non parametric tests
Assumed distribution Normal Any
Typical data Numerical Ordinal or nominal
Usual central measure Mean Median , mode
Advantages Can draw more conclusions Simplicity ; less affected by outliers
Describe one group Mean , SD Median, interquartile range
Proportion
Independent measures,2 groups Unpaired t test Chi-square test, Fisher’s test
Mann-Whitney U test
Independent measures ,>2 groups ANOVA Kruskal – Wallis test
Chi-square test
Repeated measures,2 conditions Paired t test Wilcoxon sign rank, Mc Nemar’s
Chi-square test
Repeated measures,>2 conditions ANOVA Friedman’s test
Chi-square test
Regression Simple linear regression or
Non-linear regression
Non parametric regression
10
n > 30
t
Normally
Distributed
t
Transform for t or Sign test
No
Yes
No
Yes
Difference in means or medians (ordinal or numerical measures)
1 Group
11
Number of
Groups
Independent
Groups
n > 30
t
Normally
Distributed
Equal
Variances
t
Equal n’s
t
Transform
for t or
Wilcoxon
rank-sum
Transform
for t or
Wilcoxon
rank-sum
2 Groups
Yes
Yes
No
No
Yes
Yes
No
No
Yes
Difference in means or medians (ordinal or numerical measures)
12
Independent
Groups
n > 30
t ( Paired t)
Normally
Distributed
t ( Paired t)
Transform for t
Or
Wilcoxon signed
ranks
Yes
No
No
Yes
No
13
Independent
Groups
Normally
Distributed
Number of
Factors
One-Way or
Other ANOVA
Two-Way or
Other ANOVA
Kruskal-Wallis for
1 Factor
Normally
Distributed
Repeated
Measures ANOVA
Friedman
Yes
Yes
Yes
No
No
No
1 factor
2 or
More
Factors3 or More
Groups
Difference in means or medians (ordinal or numerical measures, three or more groups)
14
Number of Groups
np and
n(1-p) > 5
z Approximation
Independent
Groups
Small Expected
frequencies
Fisher's Exact
Test
X2 or Z
Approximation
Mcnemar or K
Independent
Groups
Small Expected
Frequencies
Collapse
Categories for
X2
X2
Cochran's Q
1 Group
2 Groups
3 or More
Group
Yes
Yes
Yes
Yes
No
No
No
No
Difference in proportions (nominal measure) 15
PARAMETRIC TEST
16
t distribution and t test
• symmetric
• mean of 0, but SD > 1, the SD is related to degrees of freedom (df)
• t test:
• Corresponds to Z test
• Assumptions:
• Observations are normally distributed
• Samples must be random
• Sample size is fewer than 30
t=
𝑋 − 
𝑆𝐷
√𝑛
=
𝑋 − 
𝑆𝐸
17
𝑋=Sample Mean
=Population Mean
SD= Standard Deviation
n= sample size
SE= Standard Error
18
• Dependent t test
• Paired sample t test, within subjects or repeated measures
• Compares the means between two related groups on the same, continuous variable
• Assumptions:
• Dependent variable should be continuous variable
• Independent variable should be two categorical, related groups or matched pairs
• No significant outliers
• Distribution of differences in the dependent variable between the two related
groups should be approximately normally distributed
19
Example:
• The measurement of the systolic and diastolic blood pressures was done two consecutive
times with an interval of 10 minutes. You want to determine whether there was any
difference between those two measurements
• Null hypothesis: H0:There is no difference of the systolic blood pressure during the first
(time 0) and second measurement (time 10 minutes).
SBP 1 SBP2 D D2
164 163 1.00 1.00
164 155 9.00 81.00
156 158 -2.00 4.00
147 131 16.00 256.00
186 178 8.00 64.00
170 160 10.00 100.00
20
• Σ D = 46, Σ D2 = 506, n = 6
• Mean D = 46/6 = 7.6
21
D 𝑫 D- 𝑫 (D- 𝑫) (D- 𝑫)2
1 7.6 1-7.6 -6.6 43.56
9 7.6 9-7.6 1.4 1.96
-2 7.6 -2-7.6 -9.6 92.16
16 7.6 16-7.6 8.4 70.56
8 7.6 8-7.6 0.4 0.16
10 7.6 10-7.6 2.4 5.76
214.16
• SD =  Σ (D- 𝑫)2 /(n-1)
• SD= 214.16/5 = 42.832= 6.54
• t=
𝐷 − 
𝑆𝐷
√𝑛
• t= 7.6/(6.54/  6) = 7.6/(6.54/2.44)= 7.6/2.68=2.835
• T=2.835
• df = n – 1 = 6 – 1 = 5
• Refer to t table:
t= 2.571 at α = 0.05, df=5
• So we will reject the null hypothesis
• Conclusion: There is a significant difference of the systolic blood pressure
between the first and second measurement. The mean average of first
reading is significantly higher compared to the second reading
22
23
Independent t test
• Independent samples t test/ student’s t test/ two sample t test
• Compares the means between two unrelated groups on the same continuous,
dependent variable
• Determines whether there is a statistically significant difference between the
means of two unrelated groups
• Assumptions:
• Dependent variable should be measured on a continuous scale
• Independent variable should consist of two categorical, independent groups
• Independence of observations
• No significant outliers
• Dependent variable should be approximately normally distributed for each group of the
independent variable
• Homogeneity of variance (Levene’s test)
24
• t(n1+n2-2) =
𝑋1− 𝑋2
𝑆𝐷 𝑃
1
𝑛1+ 1
𝑛2
• SDP =
𝑛1 −1 𝑆𝐷1
2+ 𝑛2 −1 𝑆𝐷2
2
𝑛1+𝑛2 −2
• Example: Sample of size 25 was selected from healthy population, their mean
SBP =125 mm Hg with SD of 10 mm Hg . Another sample of size 17 was
selected from the population of diabetics, their mean SBP was 132 mmHg,
with SD of 12 mm Hg . Test whether there is a significant difference in mean
SBP of diabetics and healthy individual at 1% level of significance
X1 = mean of first group
X2 = mean of second group
SDP = pooled standard deviation
25
SDP =
𝑛1 −1 𝑆𝐷1
2+ 𝑛2 −1 𝑆𝐷2
2
𝑛1+𝑛2 −2
SD1 = 10
SD2 =12
State H0 H0 : 1 = 2
State H1 H1 : 1  2
Choose α α = 0.01
SDP =
25 −1 102
+ 17−1 122
25+17−2
= 10.84
t40 =
125−132
10.84 1
25+ 1
17
= 2.15
Since the computed t is smaller than critical t so there is no significant difference
between mean SBP of healthy and diabetic samples at 1 % level of significance
𝑋1 = 125
𝑋2= 132
𝑛1= 25
𝑛2= 17
26
t(n1+n2-2) =
𝑋1− 𝑋2
𝑆𝐷 𝑃
1
𝑛1+ 1
𝑛2
27
F distribution
• Distribution of ratios
• F ratio =
𝑆 𝑥
2
𝑆 𝑦
2 =
𝑙𝑎𝑟𝑔𝑒𝑟 𝑠𝑎𝑚𝑝𝑙𝑒 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒
𝑠𝑚𝑎𝑙𝑙𝑒𝑟 𝑠𝑎𝑚𝑝𝑙𝑒 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒
(𝑛 𝑥 −1 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑓 𝑓𝑟𝑒𝑒𝑑𝑜𝑚)
(𝑛 𝑦 −1 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑓 𝑓𝑟𝑒𝑒𝑑𝑜𝑚)
28
ANOVA(ANalysis Of VAriance)
• Three or more groups
• OMNIBUS test
• Assumptions:
• Dependent variable should be continuous
• Dependent variable is normally distributed in each group
• Homogeneity of variance
• Independence of observations
• No significant outliers 29
30
31
SST
(total/overall) sum
of square
SSC
(column/between/ treatment)
Sum of squares
SSE
(within/ error)
Sum of squares
SSC = ( 𝑋 - 𝑋)2
Partitioning sum of squares:
SSE =  ( X - 𝑋 )2
Variance = average squared deviation of an observation from the distribution mean
Sample variance (S2) =
∑(𝜒− 𝜇)2
𝑛−1
Sum of Squares, SS =  ( - )2
Year 1 scores Year 2 scores Year 3 scores
82 71 64
93 62 73
61 85 87
74 94 91
69 78 56
70 66 78
53 71 87
𝑋1 = 71.71 𝑋2 = 75.29 𝑋3 = 76.57
Overall mean 𝑋 = 74.52
32
Question: Is there a difference in mean scores of 3 groups of students
• SSC, dfcolumns = C – 1 MSC =
𝑆𝑆𝐶
𝑑𝑓 𝑐𝑜𝑙𝑢𝑚𝑛𝑠
=
88.67
2
= 44.33
• SSE, dferror = N – C MSE =
𝑆𝑆𝐶
𝑑𝑓𝑒𝑟𝑟𝑜𝑟
=
2812.57
18
= 156.25
• SST, dftotal = N-1 F =
𝑀𝑆𝐶
𝑀𝑆𝐸
= F =
44.33
156.25
= 0.28
MSC- Mean sum squares column
MSE- Mean sum squares error
N – Total number of subjects
C – Number of columns
33
F =
𝑑𝑓=2
𝑑𝑓=18
Fcritical = 3.55 (Fα, df c, df E = F.05, 2, 18 )
Null hypothesis for ANOVA:
H0: 1 = 2 =3 = 4…………
F-stat smaller than F-critical
FAIL TO REJECT null hypothesis
Source of variance Df SS MS F
Between (columns) 2 12.69 6.35 0.04
Within (error) 18 2812.57 156.25
Total 20 2825.26
34
35
Failure of assumptions:
• Data is not randomly distributed
• Transform data
• Non parametric test (Kruskal-Wallis H test)
• Violation of homogeneity of variance
• Welch test
• Brown and Forsythe test
• Lack of independence of data
• Very little can be done!
36
Two way ANOVA
• Mean differences between groups that have been split on two independent variable
(factors)
• Tells us if there is an interaction between two independent variables on the dependent
variable
• Assumptions:
• Dependent variable is continuous
• Independent variable - two or more categorical, independent groups
• Independence of observations
• No significant outliers
• Dependent variable - normally distributed
• Homogeneity of variances
37
38
Question: If a difference existed in insulin sensitivity depending on thyroid level or body
mass index (BMI)
4 Treatment combinations: overweight hyperthyroid subjects, overweight controls, normal
weight hyperthyroid subjects and normal weight controls
3 Questions:
A. Do differences exist between hyperthyroid subjects and controls?
B. Do differences exist between overweight and normal weight subjects?
C. Do differences exist due to neither thyroid status nor weight alone but to the
combination of factors?
39
A. Difference between Patients and Controls
B. Difference between Overweight and Normal weight subjects
C. Difference owing only to Combination of Factors
Weight
Subjects Overweight Normal
Patients 1.00 1.00
Controls 0.50 0.50
Weight
Subjects Overweight Normal
Patients 0.50 1.00
Controls 0.50 1.00
Weight
Subjects Overweight Normal
Patients 0.50 1.00
Controls 1.00 0.50
INTERACTION
No interaction: effects are additive Significant interaction: effects are
multiplicative
40
NON PARAMETRIC TEST
41
Sign test
• Used to determine if there is a median difference between paired or matched
observations
• Assumptions:
• Dependent variable is continuous
• Independent variable consists of two categorical, related groups or matched pairs
• Paired observations for each participant is independent
42
Example:
• Data:
• H0; N=8
H1; N>8 (if any observation is equal to 8, eliminate it and decrease by one)
• Test statistic: S+ = number of sample observation greater than 8 = 7
we will reject H0 if S+ is ‘sufficiently large’
• P-value: the number of observations greater than 8 is a binomial random variable,X,
and S+ is the observed value of X
• n = 10 trials, Probality (p)= 0.5
• P value= p (X ≥ 7) = 1 – P (X ≤ 6) = 1 – 0.828 = 0.172
• P value (0.172) > α = 0.05 Fail to reject Null hypothesis
2.4 15.6 14.3 11.2 9.4 3.9 11.6 8.4 12.5 6.8
- + + + + - + + + -
43
44
Wilcoxon signed rank test
• Non parametric equivalent to dependent t test
• Used when assumption of normality has been violated and use of dependent t
test is inappropriate
• To compare two sets of scores that come from the same participants
• Assumptions:
• Dependent variable is continuous or ordinal
• Independent variable should consist of two categorical, related groups or
matched pairs
• Distribution of the differences between two related groups is symmetrical
45
Candidate 1 2 3 4 5 6 7 8 9 10 11 12 13
Mock 40 65 53 70 87 42 80 63 51 82 27 71 29
Final exam 45 68 47 75 88 60 77 69 60 88 30 73 35
Find the difference
Difference 5 3 -6 -4 1 18 -3 6 9 6 3 2 6
Rearrange 1 2 3 3 3 4 5 6 6 6 6 9 18
Rank 1 2 4 4 4 6 7 9 9 9 9 12 13
Rank 1 2 3 4 5 6 7 8 9 10 11 12 13
Rank 7 4 9 6 1 13 4 9 12 9 4 2 9
T+ = 70, T- = 19
46
• Wilcoxon table
1 tail, 5%, n=13
The critical value is 21
Test statistic is 19
Test statistic < Critical value
REJECT NULL HYPOTHESIS
47
Mann-Whitney U test/ Wilcoxon Rank Sum test
• Compares median
• Assumption:
• Dependent variable should be ordinal or continuous
• Independent variable should have two categorical or continuous variable
• Independence of observation
• Two distributions of the independent variables have the same shape
48
Treatment A Treatment B
3 9
4 7
2 5
6 10
2 6
5 8
• H0 : No difference between the ranks of each treatment
• H1 : There is a difference between the ranks of each treatment
• α = 0.05
Rank Original
value
New Rank
1 2 1.5
2 2 1.5
3 3 3
4 4 4
5 5 5.5
6 5 5.5
7 6 7.5
8 6 7.5
9 7 9
10 8 10
11 9 11
12 10 12
stat = Rank sum – n(n+1)/2
A = 23 – 6(6+1)/2 = 2
B = 55 – 6(6+1)/2 = 34
stat = 2, critical = 5
stat < critical Reject null hypothesis
Treatment A Treatment B
3 11
4 9
1.5 5.5
7.5 12
1.5 7.5
5.5 10
(23) (55)
49
50
Kruskal-Wallis H test
• Rank based non parametric alternative of One way ANOVA
• OMNIBUS test
• Assumptions:
• Dependent variable is ordinal or continuous and not normally distributed
• Independent variable consist of two or more categorical, independent groups
• Independence of variance
• Distributions in each groups has the same shape
51
A Priori, or Planned Comparisons
• Multiple groups of scores, but specific comparisons have been planned prior to data
collection
• E.g. If we have 3 groups and it has been decided before data collection that comparisons
will be made between groups 1 and 2, groups 1 and 3 or groups 2 and 3
• Orthogonal t-test
• Modification:
t = 𝑋1−𝑋2
2 𝑀𝑆𝐸 𝑛
Independent
52
The Bonferroni correction
• Error rate correction formula for overcomparison
• Multiple comparison correction when several dependent or independent
statistical tests are being performed
• Alpha value is lowered to account for the number of comparisons performed
• Sets the alpha value for the entire set of ‘C’ comparisons
α = αFW  C
(New alpha to (Family wise (Number of comparisons)
correct for Type I error) error alpha)
53
e.g.
• 5 treatment groups
• Compare between following sets of groups:
• 1 to 2, 2 to 3, 3 to 4, 4 to 5, 1 to 3, 2 to 4, 3 to 5, 4 to 6, (1 & 2) to (3 & 4) and (4 to 5)
10 comparisons
• Bonferroni correction
0.05  10 = 0.005 (new critical alpha)
54
A Posteriori, or Post Hoc Comparisons
• Tukey's test
• The Scheffé post hoc test
• The Newman–Kuels post hoc test
• Dunnett's procedure
55
Tukey’s HSD Procedure
• Done if null hypothesis is rejected
• Calculates a value based upon the mean squared error, the sample size and a value from a
studentized Q range distribution
• If difference between two sample means is greater than HSD, then they are statistically
significant
Honestly significant difference
HSD = Qα, C, N-C MSE/𝑛
α = level of significance
C = number of columns/groups
N = total number of observations
n = number of observations in each group
56
Example: 5 groups with 9 observations in each
1 2 3 4 5
Mean 14.5 13.8 13.3 14.3 13.1
• HO : means are equal among all the samples
• One way ANOVA:
• F = 37.84 > Fcritical = 2.61, Reject Null hypothesis
df SS MS F
Column 4 13.32 3.33 37.84
Error 40 3.53 0.088
Total 44 16.85
57
• Q α, C, N-C = Q o.5, 5, 40 = 4.04
• HSD = Qα, C, N-C MSE/𝑛
= 4.04 0.088/9= 0.4
• Arrange means in increasing order:
• Pairs not underscored by the same line are significantly different from one another
• 1 and 4 are not significantly different from one another. 1 and 4 are significantly different
from 2, 3, 5
• 3 and 5 are not significantly different from one another
• 2 is significantly different from 3 and 5
5 3 2 4 1
Mean 13.1 13.3 13.8 14.3 14.5
58
59
CHI SQUARE TEST
• Test of significance of association between two or more qualitative variables
• Chi-square test is a very versatile statistical procedure used to test for differences in
proportions as well as an association between two variable
The assumptions made for applying chi square test are:
Random sample data
Sufficiently large sample
Normal distribution of deviation and not of data
Nominal/ordinal/interval data may be used (qualitative)
Logical/empirical basis for classifying data into nominal group
60
Chi-square is used for:
Test of proportion
Test of association
Test of goodness of fit
Chi-square test is a test of significance of association between two variables
2 =  (O – E)2
E
Where, O= observed frequencies
And E= expected frequencies
Expected frequencies are calculated as:
𝐑𝐨𝐰 𝐭𝐨𝐭𝐚𝐥 𝐗 𝐂𝐨𝐥𝐮𝐦𝐧 𝐭𝐨𝐭𝐚𝐥
𝐆𝐫𝐚𝐧𝐝 𝐭𝐨𝐭𝐚𝐥
Degree of freedom for critical 2 = (Row - 1) (Column - 1)
61
Example:
• In a study to determine the effect of heredity in a certain disease, a sample of
cases and controls was taken:
• Using 5% level of significance, test whether family history has an effect on
disease
Family history Disease
Total
Cases Controls
Positive 80 120 200
Negative 140 160 300
Total 220 280 500
62
2= (80-88)2/88 + (120-112)2/112 + (140-132)2/132 + (160-168)2/168
= 2.165
2 < 3.84
No significant association
Family history Disease
Total
Cases Controls
Positive
O
E
80
88
120
112
200
Negative
O
E
140
132
160
168
300
Total 220 280 500
63
Fisher’s exact test
 An exact test for 2 × 2 contingency tables. It is used when the sample size is too small to
use the chi-square test.
 Preferred to chi-square when two characteristics are being compared, each at two levels
because it provides the exact probability
 If any expected frequency is less than 2 or if more than 20% of the expected frequencies
are less than 5
64
65
sex Pain medication A Pain medication B Total
Male 2(2.5) 7(6.5) 9
Female 4(3.5) 9(9.5) 13
Total 6 16 22
1.Make sure no assumptions are violated.
2.State null hypothesis and alternative hypothesis
H0= there is no association between patient’s gender and pain medication used
H1=there is an association between gender and pain medication
3.State α (0.05)
4.Calculate df (NA)
5. Find critical chi square value(NA)
6.Calculate p value
Fisher’s directly calculate p value
Here p value=1 (>0.05)
Thus , there is not a statistically significant association between gender and pain
medication.
Continuity Correction (Yates' correction)
• It involves subtracting ½ from the difference between observed and expected
frequencies in the numerator of χ2 before squaring; it has the effect of making the
value for χ2 smaller
• A smaller value for χ2 means that the null hypothesis will not be rejected as often as
it is with the larger, uncorrected chi-square; that is, it is more conservative
• Thus, the risk of a type I error (rejecting the null hypothesis when it is true) is smaller
• However, the risk of a type II error (not rejecting the null hypothesis when it is false
and should be rejected) then increases
66
McNemar test
•In studies in which the outcome is a binary (yes/no) variable, researchers
may want to know whether the proportion of subjects with (or without)
the characteristic of interest changes after an intervention or the passage
of time.
•In these types of studies, we need a statistical test that is similar to the
paired t test and appropriate with nominal data.
67
Example:
• The researchers wanted to know whether changes occurred in the bowel
function of patients following cholecystectomy. They collected information
on the number of patients who had one or fewer versus more than one stool
per day :
Before
cholecystectomy
1 month after
cholecystectomy Total
≤1 >1
≤1 25 15 40
>1 0 11 11
Total 25 26 51
68
•Because 15 is larger than 3.84( at α = 0.05, critical value is 3.84), we can
reject the null hypothesis and conclude that there is a difference in the
proportion of patients having more than one stool per day before and after
cholecystectomy.
•As with the z statistic, it is possible to use a continuity correction with the
McNemar test. The correction involves subtracting 1 from the absolute value
in the numerator before squaring it.
69
70
Thank You

Más contenido relacionado

La actualidad más candente

La actualidad más candente (20)

The mann whitney u test
The mann whitney u testThe mann whitney u test
The mann whitney u test
 
Significance test
Significance testSignificance test
Significance test
 
tests of significance
tests of significancetests of significance
tests of significance
 
Student t-test
Student t-testStudent t-test
Student t-test
 
Parametric tests
Parametric  testsParametric  tests
Parametric tests
 
P value
P valueP value
P value
 
Tests of significance z &amp; t test
Tests of significance z &amp; t testTests of significance z &amp; t test
Tests of significance z &amp; t test
 
t-TEst. :D
t-TEst. :Dt-TEst. :D
t-TEst. :D
 
Kruskal Wall Test
Kruskal Wall TestKruskal Wall Test
Kruskal Wall Test
 
t test
t testt test
t test
 
Biostatistics
BiostatisticsBiostatistics
Biostatistics
 
Student's T-test, Paired T-Test, ANOVA & Proportionate Test
Student's T-test, Paired T-Test, ANOVA & Proportionate TestStudent's T-test, Paired T-Test, ANOVA & Proportionate Test
Student's T-test, Paired T-Test, ANOVA & Proportionate Test
 
P value, Power, Type 1 and 2 errors
P value, Power, Type 1 and 2 errorsP value, Power, Type 1 and 2 errors
P value, Power, Type 1 and 2 errors
 
Analysis of variance
Analysis of varianceAnalysis of variance
Analysis of variance
 
Non parametric test
Non parametric testNon parametric test
Non parametric test
 
T test
T testT test
T test
 
Mann Whitney U test
Mann Whitney U testMann Whitney U test
Mann Whitney U test
 
Non parametric test
Non parametric testNon parametric test
Non parametric test
 
Test of significance in Statistics
Test of significance in StatisticsTest of significance in Statistics
Test of significance in Statistics
 
Analysis of variance ppt @ bec doms
Analysis of variance ppt @ bec domsAnalysis of variance ppt @ bec doms
Analysis of variance ppt @ bec doms
 

Destacado

Test of significance (t-test, proportion test, chi-square test)
Test of significance (t-test, proportion test, chi-square test)Test of significance (t-test, proportion test, chi-square test)
Test of significance (t-test, proportion test, chi-square test)Ramnath Takiar
 
What's Significant? Hypothesis Testing, Effect Size, Confidence Intervals, & ...
What's Significant? Hypothesis Testing, Effect Size, Confidence Intervals, & ...What's Significant? Hypothesis Testing, Effect Size, Confidence Intervals, & ...
What's Significant? Hypothesis Testing, Effect Size, Confidence Intervals, & ...Pat Barlow
 
Medical informatics
Medical informaticsMedical informatics
Medical informaticsmigom doley
 
Descriptive epidemiology
Descriptive epidemiologyDescriptive epidemiology
Descriptive epidemiologymigom doley
 
Research methodology Chapter 6
Research methodology Chapter 6Research methodology Chapter 6
Research methodology Chapter 6Pulchowk Campus
 
Research methodology hypothesis construction
Research methodology   hypothesis constructionResearch methodology   hypothesis construction
Research methodology hypothesis constructionSami Nighaoui
 
Malimu statistical significance testing.
Malimu statistical significance testing.Malimu statistical significance testing.
Malimu statistical significance testing.Miharbi Ignasm
 
National AIDS control program
National AIDS control programNational AIDS control program
National AIDS control programmigom doley
 
Operational research in Public Health in India
Operational research in Public Health in IndiaOperational research in Public Health in India
Operational research in Public Health in IndiaDr. Dharmendra Gahwai
 
Monitoring and evaluation
Monitoring and evaluationMonitoring and evaluation
Monitoring and evaluationmigom doley
 
5. Identifying variables and constructing hypothesis
5. Identifying variables and constructing hypothesis5. Identifying variables and constructing hypothesis
5. Identifying variables and constructing hypothesisRazif Shahril
 
Levels of measurement
Levels of measurementLevels of measurement
Levels of measurementBbte Rein
 
Measurement scales
Measurement scalesMeasurement scales
Measurement scalesAiden Yeh
 

Destacado (20)

Test of significance
Test of significanceTest of significance
Test of significance
 
Test of significance (t-test, proportion test, chi-square test)
Test of significance (t-test, proportion test, chi-square test)Test of significance (t-test, proportion test, chi-square test)
Test of significance (t-test, proportion test, chi-square test)
 
What's Significant? Hypothesis Testing, Effect Size, Confidence Intervals, & ...
What's Significant? Hypothesis Testing, Effect Size, Confidence Intervals, & ...What's Significant? Hypothesis Testing, Effect Size, Confidence Intervals, & ...
What's Significant? Hypothesis Testing, Effect Size, Confidence Intervals, & ...
 
Medical informatics
Medical informaticsMedical informatics
Medical informatics
 
06 3
06 306 3
06 3
 
Descriptive epidemiology
Descriptive epidemiologyDescriptive epidemiology
Descriptive epidemiology
 
Research methodology Chapter 6
Research methodology Chapter 6Research methodology Chapter 6
Research methodology Chapter 6
 
Research methodology hypothesis construction
Research methodology   hypothesis constructionResearch methodology   hypothesis construction
Research methodology hypothesis construction
 
Malimu statistical significance testing.
Malimu statistical significance testing.Malimu statistical significance testing.
Malimu statistical significance testing.
 
National AIDS control program
National AIDS control programNational AIDS control program
National AIDS control program
 
Operational research in Public Health in India
Operational research in Public Health in IndiaOperational research in Public Health in India
Operational research in Public Health in India
 
Monitoring and evaluation
Monitoring and evaluationMonitoring and evaluation
Monitoring and evaluation
 
Leprosy
LeprosyLeprosy
Leprosy
 
Measurement levels
Measurement levelsMeasurement levels
Measurement levels
 
5. Identifying variables and constructing hypothesis
5. Identifying variables and constructing hypothesis5. Identifying variables and constructing hypothesis
5. Identifying variables and constructing hypothesis
 
Levels of Measurement
Levels of MeasurementLevels of Measurement
Levels of Measurement
 
Levels of measurement
Levels of measurementLevels of measurement
Levels of measurement
 
The Level Measurement
The Level MeasurementThe Level Measurement
The Level Measurement
 
Level Of Measurement
Level Of MeasurementLevel Of Measurement
Level Of Measurement
 
Measurement scales
Measurement scalesMeasurement scales
Measurement scales
 

Similar a Test of significance

Statistical analysis.pptx
Statistical analysis.pptxStatistical analysis.pptx
Statistical analysis.pptxChinna Chadayan
 
Parametric test - t Test, ANOVA, ANCOVA, MANOVA
Parametric test  - t Test, ANOVA, ANCOVA, MANOVAParametric test  - t Test, ANOVA, ANCOVA, MANOVA
Parametric test - t Test, ANOVA, ANCOVA, MANOVAPrincy Francis M
 
inferentialstatistics-210411214248.pdf
inferentialstatistics-210411214248.pdfinferentialstatistics-210411214248.pdf
inferentialstatistics-210411214248.pdfChenPalaruan
 
Lecture 11 Paired t test.pptx
Lecture 11 Paired t test.pptxLecture 11 Paired t test.pptx
Lecture 11 Paired t test.pptxshakirRahman10
 
Dr.Dinesh-BIOSTAT-Tests-of-significance-1-min.pdf
Dr.Dinesh-BIOSTAT-Tests-of-significance-1-min.pdfDr.Dinesh-BIOSTAT-Tests-of-significance-1-min.pdf
Dr.Dinesh-BIOSTAT-Tests-of-significance-1-min.pdfHassanMohyUdDin2
 
ANOVA_PDF.pdf biostatistics course material
ANOVA_PDF.pdf biostatistics course materialANOVA_PDF.pdf biostatistics course material
ANOVA_PDF.pdf biostatistics course materialAmanuelIbrahim
 
Statistics for Medical students
Statistics for Medical studentsStatistics for Medical students
Statistics for Medical studentsANUSWARUM
 
Tugasan kumpulan anova
Tugasan kumpulan anovaTugasan kumpulan anova
Tugasan kumpulan anovapraba karan
 
t distribution, paired and unpaired t-test
t distribution, paired and unpaired t-testt distribution, paired and unpaired t-test
t distribution, paired and unpaired t-testBPKIHS
 
descriptive statistics.pptx
descriptive statistics.pptxdescriptive statistics.pptx
descriptive statistics.pptxTeddyteddy53
 
Parametric tests seminar
Parametric tests seminarParametric tests seminar
Parametric tests seminardrdeepika87
 
2.0.statistical methods and determination of sample size
2.0.statistical methods and determination of sample size2.0.statistical methods and determination of sample size
2.0.statistical methods and determination of sample sizesalummkata1
 
NON-PARAMETRIC TESTS.pptx
NON-PARAMETRIC TESTS.pptxNON-PARAMETRIC TESTS.pptx
NON-PARAMETRIC TESTS.pptxDrLasya
 
DOE Project ANOVA Analysis Diet Type
DOE Project ANOVA Analysis Diet TypeDOE Project ANOVA Analysis Diet Type
DOE Project ANOVA Analysis Diet Typevidit jain
 
Anova one way sem 1 20142015 dk
Anova one way sem 1 20142015 dkAnova one way sem 1 20142015 dk
Anova one way sem 1 20142015 dkSyifa' Humaira
 

Similar a Test of significance (20)

Statistical analysis.pptx
Statistical analysis.pptxStatistical analysis.pptx
Statistical analysis.pptx
 
Parametric test - t Test, ANOVA, ANCOVA, MANOVA
Parametric test  - t Test, ANOVA, ANCOVA, MANOVAParametric test  - t Test, ANOVA, ANCOVA, MANOVA
Parametric test - t Test, ANOVA, ANCOVA, MANOVA
 
inferentialstatistics-210411214248.pdf
inferentialstatistics-210411214248.pdfinferentialstatistics-210411214248.pdf
inferentialstatistics-210411214248.pdf
 
Inferential statistics
Inferential statisticsInferential statistics
Inferential statistics
 
Lecture 11 Paired t test.pptx
Lecture 11 Paired t test.pptxLecture 11 Paired t test.pptx
Lecture 11 Paired t test.pptx
 
Dr.Dinesh-BIOSTAT-Tests-of-significance-1-min.pdf
Dr.Dinesh-BIOSTAT-Tests-of-significance-1-min.pdfDr.Dinesh-BIOSTAT-Tests-of-significance-1-min.pdf
Dr.Dinesh-BIOSTAT-Tests-of-significance-1-min.pdf
 
ANOVA_PDF.pdf biostatistics course material
ANOVA_PDF.pdf biostatistics course materialANOVA_PDF.pdf biostatistics course material
ANOVA_PDF.pdf biostatistics course material
 
Statistical analysis
Statistical  analysisStatistical  analysis
Statistical analysis
 
Statistics for Medical students
Statistics for Medical studentsStatistics for Medical students
Statistics for Medical students
 
Tugasan kumpulan anova
Tugasan kumpulan anovaTugasan kumpulan anova
Tugasan kumpulan anova
 
t distribution, paired and unpaired t-test
t distribution, paired and unpaired t-testt distribution, paired and unpaired t-test
t distribution, paired and unpaired t-test
 
descriptive statistics.pptx
descriptive statistics.pptxdescriptive statistics.pptx
descriptive statistics.pptx
 
Parametric tests seminar
Parametric tests seminarParametric tests seminar
Parametric tests seminar
 
2.0.statistical methods and determination of sample size
2.0.statistical methods and determination of sample size2.0.statistical methods and determination of sample size
2.0.statistical methods and determination of sample size
 
NON-PARAMETRIC TESTS.pptx
NON-PARAMETRIC TESTS.pptxNON-PARAMETRIC TESTS.pptx
NON-PARAMETRIC TESTS.pptx
 
Shovan anova main
Shovan anova mainShovan anova main
Shovan anova main
 
Comparing means
Comparing meansComparing means
Comparing means
 
DOE Project ANOVA Analysis Diet Type
DOE Project ANOVA Analysis Diet TypeDOE Project ANOVA Analysis Diet Type
DOE Project ANOVA Analysis Diet Type
 
Anova one way sem 1 20142015 dk
Anova one way sem 1 20142015 dkAnova one way sem 1 20142015 dk
Anova one way sem 1 20142015 dk
 
Parametric tests
Parametric testsParametric tests
Parametric tests
 

Más de migom doley

Más de migom doley (8)

Diabetes Mellitus
Diabetes MellitusDiabetes Mellitus
Diabetes Mellitus
 
RBSK
RBSKRBSK
RBSK
 
Essential medicine list
Essential medicine listEssential medicine list
Essential medicine list
 
Demography
DemographyDemography
Demography
 
Hypertension
Hypertension Hypertension
Hypertension
 
Poliomyelitis
Poliomyelitis Poliomyelitis
Poliomyelitis
 
Malaria
MalariaMalaria
Malaria
 
Inventory management principles
Inventory management principlesInventory management principles
Inventory management principles
 

Último

AUTONOMIC NERVOUS SYSTEM organization and functions
AUTONOMIC NERVOUS SYSTEM organization and functionsAUTONOMIC NERVOUS SYSTEM organization and functions
AUTONOMIC NERVOUS SYSTEM organization and functionsMedicoseAcademics
 
Mental health Team. Dr Senthil Thirusangu
Mental health Team. Dr Senthil ThirusanguMental health Team. Dr Senthil Thirusangu
Mental health Team. Dr Senthil Thirusangu Medical University
 
PAIN/CLASSIFICATION AND MANAGEMENT OF PAIN.pdf
PAIN/CLASSIFICATION AND MANAGEMENT OF PAIN.pdfPAIN/CLASSIFICATION AND MANAGEMENT OF PAIN.pdf
PAIN/CLASSIFICATION AND MANAGEMENT OF PAIN.pdfDolisha Warbi
 
BENIGN BREAST DISEASE
BENIGN BREAST DISEASE BENIGN BREAST DISEASE
BENIGN BREAST DISEASE Mamatha Lakka
 
SGK ĐIỆN GIẬT ĐHYHN RẤT LÀ HAY TUYỆT VỜI.pdf
SGK ĐIỆN GIẬT ĐHYHN        RẤT LÀ HAY TUYỆT VỜI.pdfSGK ĐIỆN GIẬT ĐHYHN        RẤT LÀ HAY TUYỆT VỜI.pdf
SGK ĐIỆN GIẬT ĐHYHN RẤT LÀ HAY TUYỆT VỜI.pdfHongBiThi1
 
blood bank management system project report
blood bank management system project reportblood bank management system project report
blood bank management system project reportNARMADAPETROLEUMGAS
 
Generative AI in Health Care a scoping review and a persoanl experience.
Generative AI in Health Care a scoping review and a persoanl experience.Generative AI in Health Care a scoping review and a persoanl experience.
Generative AI in Health Care a scoping review and a persoanl experience.Vaikunthan Rajaratnam
 
Different drug regularity bodies in different countries.
Different drug regularity bodies in different countries.Different drug regularity bodies in different countries.
Different drug regularity bodies in different countries.kishan singh tomar
 
EXERCISE PERFORMANCE.pptx, Lung function
EXERCISE PERFORMANCE.pptx, Lung functionEXERCISE PERFORMANCE.pptx, Lung function
EXERCISE PERFORMANCE.pptx, Lung functionkrishnareddy157915
 
Using Data Visualization in Public Health Communications
Using Data Visualization in Public Health CommunicationsUsing Data Visualization in Public Health Communications
Using Data Visualization in Public Health Communicationskatiequigley33
 
High-Performance Thin-Layer Chromatography (HPTLC)
High-Performance Thin-Layer Chromatography (HPTLC)High-Performance Thin-Layer Chromatography (HPTLC)
High-Performance Thin-Layer Chromatography (HPTLC)kishan singh tomar
 
Role of Soap based and synthetic or syndets bar
Role of  Soap based and synthetic or syndets barRole of  Soap based and synthetic or syndets bar
Role of Soap based and synthetic or syndets barmohitRahangdale
 
SGK RỐI LOẠN TOAN KIỀM ĐHYHN RẤT HAY VÀ ĐẶC SẮC.pdf
SGK RỐI LOẠN TOAN KIỀM ĐHYHN RẤT HAY VÀ ĐẶC SẮC.pdfSGK RỐI LOẠN TOAN KIỀM ĐHYHN RẤT HAY VÀ ĐẶC SẮC.pdf
SGK RỐI LOẠN TOAN KIỀM ĐHYHN RẤT HAY VÀ ĐẶC SẮC.pdfHongBiThi1
 
ANATOMICAL FAETURES OF BONES FOR NURSING STUDENTS .pptx
ANATOMICAL FAETURES OF BONES  FOR NURSING STUDENTS .pptxANATOMICAL FAETURES OF BONES  FOR NURSING STUDENTS .pptx
ANATOMICAL FAETURES OF BONES FOR NURSING STUDENTS .pptxWINCY THIRUMURUGAN
 
"Radical excision of DIE in subferile women with deep infiltrating endometrio...
"Radical excision of DIE in subferile women with deep infiltrating endometrio..."Radical excision of DIE in subferile women with deep infiltrating endometrio...
"Radical excision of DIE in subferile women with deep infiltrating endometrio...Sujoy Dasgupta
 
MedMatch: Your Health, Our Mission. Pitch deck.
MedMatch: Your Health, Our Mission. Pitch deck.MedMatch: Your Health, Our Mission. Pitch deck.
MedMatch: Your Health, Our Mission. Pitch deck.whalesdesign
 
Red Blood Cells_anemia & polycythemia.pdf
Red Blood Cells_anemia & polycythemia.pdfRed Blood Cells_anemia & polycythemia.pdf
Red Blood Cells_anemia & polycythemia.pdfMedicoseAcademics
 

Último (20)

AUTONOMIC NERVOUS SYSTEM organization and functions
AUTONOMIC NERVOUS SYSTEM organization and functionsAUTONOMIC NERVOUS SYSTEM organization and functions
AUTONOMIC NERVOUS SYSTEM organization and functions
 
Mental health Team. Dr Senthil Thirusangu
Mental health Team. Dr Senthil ThirusanguMental health Team. Dr Senthil Thirusangu
Mental health Team. Dr Senthil Thirusangu
 
PAIN/CLASSIFICATION AND MANAGEMENT OF PAIN.pdf
PAIN/CLASSIFICATION AND MANAGEMENT OF PAIN.pdfPAIN/CLASSIFICATION AND MANAGEMENT OF PAIN.pdf
PAIN/CLASSIFICATION AND MANAGEMENT OF PAIN.pdf
 
BENIGN BREAST DISEASE
BENIGN BREAST DISEASE BENIGN BREAST DISEASE
BENIGN BREAST DISEASE
 
SGK ĐIỆN GIẬT ĐHYHN RẤT LÀ HAY TUYỆT VỜI.pdf
SGK ĐIỆN GIẬT ĐHYHN        RẤT LÀ HAY TUYỆT VỜI.pdfSGK ĐIỆN GIẬT ĐHYHN        RẤT LÀ HAY TUYỆT VỜI.pdf
SGK ĐIỆN GIẬT ĐHYHN RẤT LÀ HAY TUYỆT VỜI.pdf
 
blood bank management system project report
blood bank management system project reportblood bank management system project report
blood bank management system project report
 
American College of physicians ACP high value care recommendations in rheumat...
American College of physicians ACP high value care recommendations in rheumat...American College of physicians ACP high value care recommendations in rheumat...
American College of physicians ACP high value care recommendations in rheumat...
 
Generative AI in Health Care a scoping review and a persoanl experience.
Generative AI in Health Care a scoping review and a persoanl experience.Generative AI in Health Care a scoping review and a persoanl experience.
Generative AI in Health Care a scoping review and a persoanl experience.
 
Different drug regularity bodies in different countries.
Different drug regularity bodies in different countries.Different drug regularity bodies in different countries.
Different drug regularity bodies in different countries.
 
EXERCISE PERFORMANCE.pptx, Lung function
EXERCISE PERFORMANCE.pptx, Lung functionEXERCISE PERFORMANCE.pptx, Lung function
EXERCISE PERFORMANCE.pptx, Lung function
 
Using Data Visualization in Public Health Communications
Using Data Visualization in Public Health CommunicationsUsing Data Visualization in Public Health Communications
Using Data Visualization in Public Health Communications
 
GOUT UPDATE AHMED YEHIA 2024, case based approach with application of the lat...
GOUT UPDATE AHMED YEHIA 2024, case based approach with application of the lat...GOUT UPDATE AHMED YEHIA 2024, case based approach with application of the lat...
GOUT UPDATE AHMED YEHIA 2024, case based approach with application of the lat...
 
High-Performance Thin-Layer Chromatography (HPTLC)
High-Performance Thin-Layer Chromatography (HPTLC)High-Performance Thin-Layer Chromatography (HPTLC)
High-Performance Thin-Layer Chromatography (HPTLC)
 
Role of Soap based and synthetic or syndets bar
Role of  Soap based and synthetic or syndets barRole of  Soap based and synthetic or syndets bar
Role of Soap based and synthetic or syndets bar
 
SGK RỐI LOẠN TOAN KIỀM ĐHYHN RẤT HAY VÀ ĐẶC SẮC.pdf
SGK RỐI LOẠN TOAN KIỀM ĐHYHN RẤT HAY VÀ ĐẶC SẮC.pdfSGK RỐI LOẠN TOAN KIỀM ĐHYHN RẤT HAY VÀ ĐẶC SẮC.pdf
SGK RỐI LOẠN TOAN KIỀM ĐHYHN RẤT HAY VÀ ĐẶC SẮC.pdf
 
ANATOMICAL FAETURES OF BONES FOR NURSING STUDENTS .pptx
ANATOMICAL FAETURES OF BONES  FOR NURSING STUDENTS .pptxANATOMICAL FAETURES OF BONES  FOR NURSING STUDENTS .pptx
ANATOMICAL FAETURES OF BONES FOR NURSING STUDENTS .pptx
 
Biologic therapy ice breaking in rheumatology, Case based approach with appli...
Biologic therapy ice breaking in rheumatology, Case based approach with appli...Biologic therapy ice breaking in rheumatology, Case based approach with appli...
Biologic therapy ice breaking in rheumatology, Case based approach with appli...
 
"Radical excision of DIE in subferile women with deep infiltrating endometrio...
"Radical excision of DIE in subferile women with deep infiltrating endometrio..."Radical excision of DIE in subferile women with deep infiltrating endometrio...
"Radical excision of DIE in subferile women with deep infiltrating endometrio...
 
MedMatch: Your Health, Our Mission. Pitch deck.
MedMatch: Your Health, Our Mission. Pitch deck.MedMatch: Your Health, Our Mission. Pitch deck.
MedMatch: Your Health, Our Mission. Pitch deck.
 
Red Blood Cells_anemia & polycythemia.pdf
Red Blood Cells_anemia & polycythemia.pdfRed Blood Cells_anemia & polycythemia.pdf
Red Blood Cells_anemia & polycythemia.pdf
 

Test of significance

  • 1. TESTS OF SIGNIFICANCE Moderator: Dr. S.K.Bhasin Presenters: Migom Parnava Kartikey 1
  • 2. Probability distribution 1. Normal distribution 2. Binomial distribution 3. Poissons distribution 2
  • 3. Normal curve • Gaussian Distribution • Continuous • Bell shaped • Symmetrical • Mean, Mode and Median coincide 3
  • 4. Confidence level and Confidence limits 4
  • 5. Z distribution • Z transformation: Z= observation – mean = X -  SD  5
  • 6. • P value: • The probability of observing a result as extreme as or more extreme than the one actually observed from chance alone (i.e., if the null hypothesis is true) • Power of a test: • Probability that a study or a trial will be able to detect a specified difference • Power= 1- β 6
  • 7. Scales of measurement • Nominal scales • E.g. outcome of surgical procedure • Ordinal scales • E.g. APGAR score, tumour staging • Numerical scale • E.g. Age, weight, number of fractures 7
  • 8. Measures of central tendency • Mean • Median • Mode Some important concepts: • Standard deviation • Variance • Standard error of mean 8
  • 9. TESTS OF STATISTICAL SIGNIFICANCE • Procedure for comparing observed data with a claim (hypothesis) whose truth we want to assess • Appropriate test of significance: • Data • Sample • Purpose 9
  • 10. Parametric tests Non parametric tests Assumed distribution Normal Any Typical data Numerical Ordinal or nominal Usual central measure Mean Median , mode Advantages Can draw more conclusions Simplicity ; less affected by outliers Describe one group Mean , SD Median, interquartile range Proportion Independent measures,2 groups Unpaired t test Chi-square test, Fisher’s test Mann-Whitney U test Independent measures ,>2 groups ANOVA Kruskal – Wallis test Chi-square test Repeated measures,2 conditions Paired t test Wilcoxon sign rank, Mc Nemar’s Chi-square test Repeated measures,>2 conditions ANOVA Friedman’s test Chi-square test Regression Simple linear regression or Non-linear regression Non parametric regression 10
  • 11. n > 30 t Normally Distributed t Transform for t or Sign test No Yes No Yes Difference in means or medians (ordinal or numerical measures) 1 Group 11
  • 12. Number of Groups Independent Groups n > 30 t Normally Distributed Equal Variances t Equal n’s t Transform for t or Wilcoxon rank-sum Transform for t or Wilcoxon rank-sum 2 Groups Yes Yes No No Yes Yes No No Yes Difference in means or medians (ordinal or numerical measures) 12
  • 13. Independent Groups n > 30 t ( Paired t) Normally Distributed t ( Paired t) Transform for t Or Wilcoxon signed ranks Yes No No Yes No 13
  • 14. Independent Groups Normally Distributed Number of Factors One-Way or Other ANOVA Two-Way or Other ANOVA Kruskal-Wallis for 1 Factor Normally Distributed Repeated Measures ANOVA Friedman Yes Yes Yes No No No 1 factor 2 or More Factors3 or More Groups Difference in means or medians (ordinal or numerical measures, three or more groups) 14
  • 15. Number of Groups np and n(1-p) > 5 z Approximation Independent Groups Small Expected frequencies Fisher's Exact Test X2 or Z Approximation Mcnemar or K Independent Groups Small Expected Frequencies Collapse Categories for X2 X2 Cochran's Q 1 Group 2 Groups 3 or More Group Yes Yes Yes Yes No No No No Difference in proportions (nominal measure) 15
  • 17. t distribution and t test • symmetric • mean of 0, but SD > 1, the SD is related to degrees of freedom (df) • t test: • Corresponds to Z test • Assumptions: • Observations are normally distributed • Samples must be random • Sample size is fewer than 30 t= 𝑋 −  𝑆𝐷 √𝑛 = 𝑋 −  𝑆𝐸 17 𝑋=Sample Mean =Population Mean SD= Standard Deviation n= sample size SE= Standard Error
  • 18. 18
  • 19. • Dependent t test • Paired sample t test, within subjects or repeated measures • Compares the means between two related groups on the same, continuous variable • Assumptions: • Dependent variable should be continuous variable • Independent variable should be two categorical, related groups or matched pairs • No significant outliers • Distribution of differences in the dependent variable between the two related groups should be approximately normally distributed 19
  • 20. Example: • The measurement of the systolic and diastolic blood pressures was done two consecutive times with an interval of 10 minutes. You want to determine whether there was any difference between those two measurements • Null hypothesis: H0:There is no difference of the systolic blood pressure during the first (time 0) and second measurement (time 10 minutes). SBP 1 SBP2 D D2 164 163 1.00 1.00 164 155 9.00 81.00 156 158 -2.00 4.00 147 131 16.00 256.00 186 178 8.00 64.00 170 160 10.00 100.00 20
  • 21. • Σ D = 46, Σ D2 = 506, n = 6 • Mean D = 46/6 = 7.6 21 D 𝑫 D- 𝑫 (D- 𝑫) (D- 𝑫)2 1 7.6 1-7.6 -6.6 43.56 9 7.6 9-7.6 1.4 1.96 -2 7.6 -2-7.6 -9.6 92.16 16 7.6 16-7.6 8.4 70.56 8 7.6 8-7.6 0.4 0.16 10 7.6 10-7.6 2.4 5.76 214.16
  • 22. • SD =  Σ (D- 𝑫)2 /(n-1) • SD= 214.16/5 = 42.832= 6.54 • t= 𝐷 −  𝑆𝐷 √𝑛 • t= 7.6/(6.54/  6) = 7.6/(6.54/2.44)= 7.6/2.68=2.835 • T=2.835 • df = n – 1 = 6 – 1 = 5 • Refer to t table: t= 2.571 at α = 0.05, df=5 • So we will reject the null hypothesis • Conclusion: There is a significant difference of the systolic blood pressure between the first and second measurement. The mean average of first reading is significantly higher compared to the second reading 22
  • 23. 23
  • 24. Independent t test • Independent samples t test/ student’s t test/ two sample t test • Compares the means between two unrelated groups on the same continuous, dependent variable • Determines whether there is a statistically significant difference between the means of two unrelated groups • Assumptions: • Dependent variable should be measured on a continuous scale • Independent variable should consist of two categorical, independent groups • Independence of observations • No significant outliers • Dependent variable should be approximately normally distributed for each group of the independent variable • Homogeneity of variance (Levene’s test) 24
  • 25. • t(n1+n2-2) = 𝑋1− 𝑋2 𝑆𝐷 𝑃 1 𝑛1+ 1 𝑛2 • SDP = 𝑛1 −1 𝑆𝐷1 2+ 𝑛2 −1 𝑆𝐷2 2 𝑛1+𝑛2 −2 • Example: Sample of size 25 was selected from healthy population, their mean SBP =125 mm Hg with SD of 10 mm Hg . Another sample of size 17 was selected from the population of diabetics, their mean SBP was 132 mmHg, with SD of 12 mm Hg . Test whether there is a significant difference in mean SBP of diabetics and healthy individual at 1% level of significance X1 = mean of first group X2 = mean of second group SDP = pooled standard deviation 25
  • 26. SDP = 𝑛1 −1 𝑆𝐷1 2+ 𝑛2 −1 𝑆𝐷2 2 𝑛1+𝑛2 −2 SD1 = 10 SD2 =12 State H0 H0 : 1 = 2 State H1 H1 : 1  2 Choose α α = 0.01 SDP = 25 −1 102 + 17−1 122 25+17−2 = 10.84 t40 = 125−132 10.84 1 25+ 1 17 = 2.15 Since the computed t is smaller than critical t so there is no significant difference between mean SBP of healthy and diabetic samples at 1 % level of significance 𝑋1 = 125 𝑋2= 132 𝑛1= 25 𝑛2= 17 26 t(n1+n2-2) = 𝑋1− 𝑋2 𝑆𝐷 𝑃 1 𝑛1+ 1 𝑛2
  • 27. 27
  • 28. F distribution • Distribution of ratios • F ratio = 𝑆 𝑥 2 𝑆 𝑦 2 = 𝑙𝑎𝑟𝑔𝑒𝑟 𝑠𝑎𝑚𝑝𝑙𝑒 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑠𝑚𝑎𝑙𝑙𝑒𝑟 𝑠𝑎𝑚𝑝𝑙𝑒 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 (𝑛 𝑥 −1 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑓 𝑓𝑟𝑒𝑒𝑑𝑜𝑚) (𝑛 𝑦 −1 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑓 𝑓𝑟𝑒𝑒𝑑𝑜𝑚) 28
  • 29. ANOVA(ANalysis Of VAriance) • Three or more groups • OMNIBUS test • Assumptions: • Dependent variable should be continuous • Dependent variable is normally distributed in each group • Homogeneity of variance • Independence of observations • No significant outliers 29
  • 30. 30
  • 31. 31 SST (total/overall) sum of square SSC (column/between/ treatment) Sum of squares SSE (within/ error) Sum of squares SSC = ( 𝑋 - 𝑋)2 Partitioning sum of squares: SSE =  ( X - 𝑋 )2 Variance = average squared deviation of an observation from the distribution mean Sample variance (S2) = ∑(𝜒− 𝜇)2 𝑛−1 Sum of Squares, SS =  ( - )2
  • 32. Year 1 scores Year 2 scores Year 3 scores 82 71 64 93 62 73 61 85 87 74 94 91 69 78 56 70 66 78 53 71 87 𝑋1 = 71.71 𝑋2 = 75.29 𝑋3 = 76.57 Overall mean 𝑋 = 74.52 32 Question: Is there a difference in mean scores of 3 groups of students
  • 33. • SSC, dfcolumns = C – 1 MSC = 𝑆𝑆𝐶 𝑑𝑓 𝑐𝑜𝑙𝑢𝑚𝑛𝑠 = 88.67 2 = 44.33 • SSE, dferror = N – C MSE = 𝑆𝑆𝐶 𝑑𝑓𝑒𝑟𝑟𝑜𝑟 = 2812.57 18 = 156.25 • SST, dftotal = N-1 F = 𝑀𝑆𝐶 𝑀𝑆𝐸 = F = 44.33 156.25 = 0.28 MSC- Mean sum squares column MSE- Mean sum squares error N – Total number of subjects C – Number of columns 33
  • 34. F = 𝑑𝑓=2 𝑑𝑓=18 Fcritical = 3.55 (Fα, df c, df E = F.05, 2, 18 ) Null hypothesis for ANOVA: H0: 1 = 2 =3 = 4………… F-stat smaller than F-critical FAIL TO REJECT null hypothesis Source of variance Df SS MS F Between (columns) 2 12.69 6.35 0.04 Within (error) 18 2812.57 156.25 Total 20 2825.26 34
  • 35. 35
  • 36. Failure of assumptions: • Data is not randomly distributed • Transform data • Non parametric test (Kruskal-Wallis H test) • Violation of homogeneity of variance • Welch test • Brown and Forsythe test • Lack of independence of data • Very little can be done! 36
  • 37. Two way ANOVA • Mean differences between groups that have been split on two independent variable (factors) • Tells us if there is an interaction between two independent variables on the dependent variable • Assumptions: • Dependent variable is continuous • Independent variable - two or more categorical, independent groups • Independence of observations • No significant outliers • Dependent variable - normally distributed • Homogeneity of variances 37
  • 38. 38 Question: If a difference existed in insulin sensitivity depending on thyroid level or body mass index (BMI) 4 Treatment combinations: overweight hyperthyroid subjects, overweight controls, normal weight hyperthyroid subjects and normal weight controls 3 Questions: A. Do differences exist between hyperthyroid subjects and controls? B. Do differences exist between overweight and normal weight subjects? C. Do differences exist due to neither thyroid status nor weight alone but to the combination of factors?
  • 39. 39 A. Difference between Patients and Controls B. Difference between Overweight and Normal weight subjects C. Difference owing only to Combination of Factors Weight Subjects Overweight Normal Patients 1.00 1.00 Controls 0.50 0.50 Weight Subjects Overweight Normal Patients 0.50 1.00 Controls 0.50 1.00 Weight Subjects Overweight Normal Patients 0.50 1.00 Controls 1.00 0.50
  • 40. INTERACTION No interaction: effects are additive Significant interaction: effects are multiplicative 40
  • 42. Sign test • Used to determine if there is a median difference between paired or matched observations • Assumptions: • Dependent variable is continuous • Independent variable consists of two categorical, related groups or matched pairs • Paired observations for each participant is independent 42
  • 43. Example: • Data: • H0; N=8 H1; N>8 (if any observation is equal to 8, eliminate it and decrease by one) • Test statistic: S+ = number of sample observation greater than 8 = 7 we will reject H0 if S+ is ‘sufficiently large’ • P-value: the number of observations greater than 8 is a binomial random variable,X, and S+ is the observed value of X • n = 10 trials, Probality (p)= 0.5 • P value= p (X ≥ 7) = 1 – P (X ≤ 6) = 1 – 0.828 = 0.172 • P value (0.172) > α = 0.05 Fail to reject Null hypothesis 2.4 15.6 14.3 11.2 9.4 3.9 11.6 8.4 12.5 6.8 - + + + + - + + + - 43
  • 44. 44
  • 45. Wilcoxon signed rank test • Non parametric equivalent to dependent t test • Used when assumption of normality has been violated and use of dependent t test is inappropriate • To compare two sets of scores that come from the same participants • Assumptions: • Dependent variable is continuous or ordinal • Independent variable should consist of two categorical, related groups or matched pairs • Distribution of the differences between two related groups is symmetrical 45
  • 46. Candidate 1 2 3 4 5 6 7 8 9 10 11 12 13 Mock 40 65 53 70 87 42 80 63 51 82 27 71 29 Final exam 45 68 47 75 88 60 77 69 60 88 30 73 35 Find the difference Difference 5 3 -6 -4 1 18 -3 6 9 6 3 2 6 Rearrange 1 2 3 3 3 4 5 6 6 6 6 9 18 Rank 1 2 4 4 4 6 7 9 9 9 9 12 13 Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 Rank 7 4 9 6 1 13 4 9 12 9 4 2 9 T+ = 70, T- = 19 46
  • 47. • Wilcoxon table 1 tail, 5%, n=13 The critical value is 21 Test statistic is 19 Test statistic < Critical value REJECT NULL HYPOTHESIS 47
  • 48. Mann-Whitney U test/ Wilcoxon Rank Sum test • Compares median • Assumption: • Dependent variable should be ordinal or continuous • Independent variable should have two categorical or continuous variable • Independence of observation • Two distributions of the independent variables have the same shape 48
  • 49. Treatment A Treatment B 3 9 4 7 2 5 6 10 2 6 5 8 • H0 : No difference between the ranks of each treatment • H1 : There is a difference between the ranks of each treatment • α = 0.05 Rank Original value New Rank 1 2 1.5 2 2 1.5 3 3 3 4 4 4 5 5 5.5 6 5 5.5 7 6 7.5 8 6 7.5 9 7 9 10 8 10 11 9 11 12 10 12 stat = Rank sum – n(n+1)/2 A = 23 – 6(6+1)/2 = 2 B = 55 – 6(6+1)/2 = 34 stat = 2, critical = 5 stat < critical Reject null hypothesis Treatment A Treatment B 3 11 4 9 1.5 5.5 7.5 12 1.5 7.5 5.5 10 (23) (55) 49
  • 50. 50
  • 51. Kruskal-Wallis H test • Rank based non parametric alternative of One way ANOVA • OMNIBUS test • Assumptions: • Dependent variable is ordinal or continuous and not normally distributed • Independent variable consist of two or more categorical, independent groups • Independence of variance • Distributions in each groups has the same shape 51
  • 52. A Priori, or Planned Comparisons • Multiple groups of scores, but specific comparisons have been planned prior to data collection • E.g. If we have 3 groups and it has been decided before data collection that comparisons will be made between groups 1 and 2, groups 1 and 3 or groups 2 and 3 • Orthogonal t-test • Modification: t = 𝑋1−𝑋2 2 𝑀𝑆𝐸 𝑛 Independent 52
  • 53. The Bonferroni correction • Error rate correction formula for overcomparison • Multiple comparison correction when several dependent or independent statistical tests are being performed • Alpha value is lowered to account for the number of comparisons performed • Sets the alpha value for the entire set of ‘C’ comparisons α = αFW  C (New alpha to (Family wise (Number of comparisons) correct for Type I error) error alpha) 53
  • 54. e.g. • 5 treatment groups • Compare between following sets of groups: • 1 to 2, 2 to 3, 3 to 4, 4 to 5, 1 to 3, 2 to 4, 3 to 5, 4 to 6, (1 & 2) to (3 & 4) and (4 to 5) 10 comparisons • Bonferroni correction 0.05  10 = 0.005 (new critical alpha) 54
  • 55. A Posteriori, or Post Hoc Comparisons • Tukey's test • The Scheffé post hoc test • The Newman–Kuels post hoc test • Dunnett's procedure 55
  • 56. Tukey’s HSD Procedure • Done if null hypothesis is rejected • Calculates a value based upon the mean squared error, the sample size and a value from a studentized Q range distribution • If difference between two sample means is greater than HSD, then they are statistically significant Honestly significant difference HSD = Qα, C, N-C MSE/𝑛 α = level of significance C = number of columns/groups N = total number of observations n = number of observations in each group 56
  • 57. Example: 5 groups with 9 observations in each 1 2 3 4 5 Mean 14.5 13.8 13.3 14.3 13.1 • HO : means are equal among all the samples • One way ANOVA: • F = 37.84 > Fcritical = 2.61, Reject Null hypothesis df SS MS F Column 4 13.32 3.33 37.84 Error 40 3.53 0.088 Total 44 16.85 57
  • 58. • Q α, C, N-C = Q o.5, 5, 40 = 4.04 • HSD = Qα, C, N-C MSE/𝑛 = 4.04 0.088/9= 0.4 • Arrange means in increasing order: • Pairs not underscored by the same line are significantly different from one another • 1 and 4 are not significantly different from one another. 1 and 4 are significantly different from 2, 3, 5 • 3 and 5 are not significantly different from one another • 2 is significantly different from 3 and 5 5 3 2 4 1 Mean 13.1 13.3 13.8 14.3 14.5 58
  • 59. 59
  • 60. CHI SQUARE TEST • Test of significance of association between two or more qualitative variables • Chi-square test is a very versatile statistical procedure used to test for differences in proportions as well as an association between two variable The assumptions made for applying chi square test are: Random sample data Sufficiently large sample Normal distribution of deviation and not of data Nominal/ordinal/interval data may be used (qualitative) Logical/empirical basis for classifying data into nominal group 60
  • 61. Chi-square is used for: Test of proportion Test of association Test of goodness of fit Chi-square test is a test of significance of association between two variables 2 =  (O – E)2 E Where, O= observed frequencies And E= expected frequencies Expected frequencies are calculated as: 𝐑𝐨𝐰 𝐭𝐨𝐭𝐚𝐥 𝐗 𝐂𝐨𝐥𝐮𝐦𝐧 𝐭𝐨𝐭𝐚𝐥 𝐆𝐫𝐚𝐧𝐝 𝐭𝐨𝐭𝐚𝐥 Degree of freedom for critical 2 = (Row - 1) (Column - 1) 61
  • 62. Example: • In a study to determine the effect of heredity in a certain disease, a sample of cases and controls was taken: • Using 5% level of significance, test whether family history has an effect on disease Family history Disease Total Cases Controls Positive 80 120 200 Negative 140 160 300 Total 220 280 500 62
  • 63. 2= (80-88)2/88 + (120-112)2/112 + (140-132)2/132 + (160-168)2/168 = 2.165 2 < 3.84 No significant association Family history Disease Total Cases Controls Positive O E 80 88 120 112 200 Negative O E 140 132 160 168 300 Total 220 280 500 63
  • 64. Fisher’s exact test  An exact test for 2 × 2 contingency tables. It is used when the sample size is too small to use the chi-square test.  Preferred to chi-square when two characteristics are being compared, each at two levels because it provides the exact probability  If any expected frequency is less than 2 or if more than 20% of the expected frequencies are less than 5 64
  • 65. 65 sex Pain medication A Pain medication B Total Male 2(2.5) 7(6.5) 9 Female 4(3.5) 9(9.5) 13 Total 6 16 22 1.Make sure no assumptions are violated. 2.State null hypothesis and alternative hypothesis H0= there is no association between patient’s gender and pain medication used H1=there is an association between gender and pain medication 3.State α (0.05) 4.Calculate df (NA) 5. Find critical chi square value(NA) 6.Calculate p value Fisher’s directly calculate p value Here p value=1 (>0.05) Thus , there is not a statistically significant association between gender and pain medication.
  • 66. Continuity Correction (Yates' correction) • It involves subtracting ½ from the difference between observed and expected frequencies in the numerator of χ2 before squaring; it has the effect of making the value for χ2 smaller • A smaller value for χ2 means that the null hypothesis will not be rejected as often as it is with the larger, uncorrected chi-square; that is, it is more conservative • Thus, the risk of a type I error (rejecting the null hypothesis when it is true) is smaller • However, the risk of a type II error (not rejecting the null hypothesis when it is false and should be rejected) then increases 66
  • 67. McNemar test •In studies in which the outcome is a binary (yes/no) variable, researchers may want to know whether the proportion of subjects with (or without) the characteristic of interest changes after an intervention or the passage of time. •In these types of studies, we need a statistical test that is similar to the paired t test and appropriate with nominal data. 67
  • 68. Example: • The researchers wanted to know whether changes occurred in the bowel function of patients following cholecystectomy. They collected information on the number of patients who had one or fewer versus more than one stool per day : Before cholecystectomy 1 month after cholecystectomy Total ≤1 >1 ≤1 25 15 40 >1 0 11 11 Total 25 26 51 68
  • 69. •Because 15 is larger than 3.84( at α = 0.05, critical value is 3.84), we can reject the null hypothesis and conclude that there is a difference in the proportion of patients having more than one stool per day before and after cholecystectomy. •As with the z statistic, it is possible to use a continuity correction with the McNemar test. The correction involves subtracting 1 from the absolute value in the numerator before squaring it. 69