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Statistical Methods in Ayurveda – A Primer
Dr. Benil.P.B MD(Ay)Associate Professor
VPSV Ayurveda College Kottakkal
P - Value•Misunderstood•Misinterpreted •Miscalculated
Probability
• Probability = Chance
Understanding Mean
Understanding Standard deviationA
B
CD
E
F
H
Measurement of Probability
Q1. What scales of measurement has been used?
Q2. Which hypothesis is to be tested? Q3. Are the samples independent or
dependent? Q4. How many sets of measures are involved?
Q1. What scales of measurements have been used?
• Measurement is assigning numbers to observations.
• Scales of measurement are of three types
–Nominal scale–Ordinal scale and –Interval scale.
Whether Normally Distributed or Not?
• The normally distributed measures will conform to a normal curve
• If we plot a graph with values of the measurement presented in the X axis and frequency of occurrence presented in the Y axis.
Normal Curve
Normal Curve - Properties
Q2. Which hypothesis has been tested?
• There are Two types of hypotheses; • Hypothesis of difference • Hypothesis of association.
Hypothesis of Difference
• States that the difference that is shown in the results obtained from the samples are also different in the larger populations from which the samples came.
Hypothesis of Association
• states that the relationship of the two (or more) sets of outcome that we see in the results obtained from the sample is also present in the larger populations from which the sample came.
Q3. Are the samples independent or dependent?
• Applicable only if the hypothesis of difference is being tested
• One sample influences the selection of the other – dependent sample.
• One sample do not influence the selection of the other - independent samples
Q4. How many sets of measures are involved?
ONE SAMPLE TRIAL CONTROL
GROUP I
GROUP II
GROUP III
Testing of HypothesisDATA
NORMALLY DISTRIBUTED
NOT NORMALLY DISTRIBUTED
PARAMETRIC TESTS
NON-PARAMETRIC
TESTS
Parametric Tests
• Comparison of Means
–t Test•One sample t- Test•Unpaired sample t-Test•Paired sample t-Test
One Sample t - Test
• If an investigator measures a variable in a single group of subjects
• To determine whether the mean for the sample differs from the population value.
Example – One Sample t - Test
• A study was conducted to assess the prevalence of Pandu in Garbhini.
• Haemoglobin concentration below 11 warrants Iron supplementation.
• Whether this sample requires Iron supplementation
• Perform a One sample t - Test
One Sample t - TestSL NO. Hb
Concentration1 11.82 12.43 10.64 14.55 11.46 13.77 14.18 12.89 10.9
10 11.211 12.912 12.413 13.814 13.215 11.9
POPULATION VALUE: 11 mg/dl
t = 1.627P = 0.126
Unpaired sample t - Tests
POPULATION
SAMPLE - I SAMPLE - II
Example – Unpaired sample t – Test
• Comparison of the efficacy of Triphala guggulu and Varanadi gana vati in reducing Total Cholesterol levels in patients with Dyslipidemia.
Unpaired Sample t - Test
SL NO.GROUP A TRIPHALA GUGGULU
GROUP B VARANADI GANA VATI
1 198 2232 203 1923 149 2024 174 2035 221 1956 186 1857 154 1788 165 1889 212 222
10 185 249MEAN 184.7 203.7SD 24.3 21.7
P = 0.056
t = 2.0418
Paired sample t - Test
• The dependent t-test for paired samples is used when the samples are paired.
• Each individual observation of one sample has a unique corresponding member in the other sample.
Paired sample t - Test
• The effect of Pavanamuktasana in reducing Abdominal circumference in Obese persons
Dependent sample t - Test
p = 0.01t = 3.44
SL NO
ABDOMINAL CIRCUMFERENCE
BEFORE INTERVENTION
AFTER INTERVENTION
1 95 90
2 109 99
3 123 103
4 98 99
5 99 94
6 108 98
7 116 90
8 108 100
9 96 90
10 110 100
MEAN 106.2 96.3
SD 9.16 4.88
Analysis of Variance
• Test that evaluate differences between three or more means
• Developed by Fischer• Fischer Test (F – Test)
• Considers Variances rather than Mean
ANalysis Of VAriance
ANOVA
ANOVA
• ANOVA – Single Factor
–One Way ANOVA• ANOVA – Two Factor
–Two Way ANOVA
Example – One Way Anova
• Compare the efficacy of three formulations Yogaraja guggulu, .Kaisora guggulu and Punarnnava guggulu in relieving pain from Sandhigata vata.
One Way ANOVASL NO
TREATMENT OPTIONS
YOGARAJA GUGGULU
KAISORA GUGGULU
PUNARNAVA GUGGULU
1 4 5 12 3 4 13 3 4 14 2 4 35 5 3 56 4 2 37 2 5 18 3 4 29 2 3 2
10 2 2 2MEAN 3 3.6 2.1SD 1.05 1.07 1.29
p = 0.02F = 4.35
ANOVA Output
ANOVASource of Variation SS df MS F P-value F crit
Between Groups 11.40 2.00 5.70 4.36 0.02 3.35Within Groups 35.30 27.00 1.31
Total 46.70 29.00
Two way ANOVA
SL NO SEXTREATMENT OPTIONS
YOGARAJA GUGGULU
KAISORA GUGGULU
PUNARNAVA GUGGULU
1
FEMALE
4 5 12 3 4 13 3 4 14 2 4 35 5 3 56
MALE
4 2 37 2 5 18 3 4 29 2 3 2
10 2 2 2MEAN 3 3.6 2.1SD 1.05 1.07 1.29
Post hoc Comparisons
• Common Post-hoc Tests• Tuckey Kramer Test• Dunnet’s Test• Bonferroni Test• Duncan Test
Example - Tuckey Kramer Test
• Comparison of the efficacy of FOUR Antipyretic formulations – Vettumaran Gutika, Jwarnkusha Rasa, Seetajwarari rasa abd Jwaraghni Gutika
Tuckey Kramer TestSL NO
ANTIPYRETIC DRUGS
VETTUMARAN JWARANKUSHA SEETAJWARARI JARAGHNI VATI
1 5 3 2 32 5 4 1 33 2 2 1 24 4 4 4 25 3 5 5 56 3 6 5 47 2 2 2 38 6 1 2 39 5 1 2 4
10 4 4 6 4MEAN 3.9 3.2 3 3.3
SD 1.4 1.7 1.8 0.9
ANOVA
Source of Variation SS df MS F P-value F crit
Between Groups 19.8 3 6.6 2.876 0.04 2.866
Within Groups 82.6 36 2.294
Total 102.4 39
SL NO COMPARISON MD P-VALUE
1 VETTUMARAN X JWARANKUSHA 0.7 P < 0.05
2 VETTUMARAN X SEETAJWARARI 1.5 P < 0.01
3 VETTUMARAN X JWARAGHNI VATI 1.8 P < 0.01
4 JWARANKUSHA X SEETAJWARARI 0.8 P < 0.05
5 JWARANKUSHA X JWARAGHNI VATI 1.1 P < 0.01
6 SEETAJWARARI X JWARAGHNI VATI 0.3 P > 0.05
VETTUMARAN JWARANKUSHA SEETAJWARARI JARAGHNI VATI0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
4.3
3.6
2.82.5
Example – Dunnet’s Test
• Comparison of THREE doses of Kiratatiktha (Andrographis paniculata) extract in Carbon tetrachloride induced Hepatotoxicity in Male Wistar Rats.
SL NO
SERUM SGOT LEVELS
STANDARD (SYLIMARIN)
CONTROL (CCl4 ALONE)
LOW DOSE (CCl4 + APE 125 mg/Kg)
MEDIUM DOSE (CCl4 + APE 250 mg/Kg)
HIGH DOSE (CCl4 + APE 500 mg/Kg)
1 32 245 130 113 43
2 28 198 124 99 42
3 35 104 111 94 39
4 12 173 128 107 31
5 23 112 132 116 24
6 19 186 122 93 42
7 29 220 107 105 41
8 32 204 112 92 36
9 35 263 128 101 40
10 42 288 132 111 12
MEAN 28.7 199.3 122.6 103.1 35
SD 8.7 59.7 9.3 8.7 10.0
ANOVA
Source of Variation SS df MS F P-value F crit
Between Groups 196640.1 4 49160 63.05 0.001 2.578
Within Groups 35085.5 45 779.6
Total 231725.6 49
SL NO COMPARISON MD P-VALUE
1 STANDARD X CONTROL 170.6 p < 0.001
2 STANDARD X LOW DOSE 93.9 p < 0.01
3 STANDARD X MEDIUM DOSE 74.4 P < 0.01
4 STANDARD X HIGH DOSE 6.3 P > 0.05
STANDARD CONTROL LOW DOSE MEDIUM DOSE HIGH DOSE 0
50
100
150
200
250
28.7
199.3
122.6
103.1
35
Non-Parametric Tests
• Chi-Square Test• Tests the difference between observed
frequencies and the frequencies expected under certain assumptions
Types of Chi-Square Test
• Chi-Square for Association• Chi-Square for Goodness of fit• Chi-Square for Independence
Chi-Squared Test of Association
• Comparison of two factors to determine relationship between them.
• Compare the observed frequencies with the expected frequencies
Chi-square test for goodness-of-fit
• To determine whether a set of frequencies “fits” with a hypothesized set of frequencies or proportions”.
• A Chi-square goodness-of-fit test is like to a one-sample t-test. It determines if a sample is similar to, and representative of, a population
Chi square Test of Independence
• A test of independence is a two variable Chi-square test.
• To determine whether a variable is related to—or independent of—the second variable”.
Example – Chi test Association
• A Case-Control Study to assess the Association between Mootra vegadharana and Dysmenorrhoea.
SL NO:AGE OF
MENARCHE VEGADHARANA PRAKRUTI DYSMENORRHOEA VAS1 14 1 1 1 62 12 0 3 1 83 9 1 2 0 04 13 1 2 0 05 12 0 1 1 96 11 1 1 1 47 11 1 3 1 68 10 1 3 1 49 12 0 2 0 0
10 14 0 1 0 0
Example – Chi square
DYSMENORRHOEA
PRESENT ABSENT
VEGADHARANA
PRESENT 5 4 8
ABSENT 1 2 4
6 6 24
p = 0.008Chi = 6.668
ODD'S RATIO = 2.5
Example – Chi square
DYSMENORRHOEA
PRESENT ABSENT
MENARCHE
BEFORE 12 3 1 4
AFTER 12 3 3 6
6 4 20
ODD'S RATIO = 3
p = 0.010Chi = 6.250
Mann-Whitney U Test
• Non-parametric counterpart of Independent samples t – Test.
Wilcoxon’s Signed Rank Test
• Non-parametric counterpart of paired sample t – Test
Kruskal- Wallis Test
• Non-parametric counterpart of One way Anova.
Tests for Normality
• Assumption of Normality–Shapiro-Wilk W test–Anderson-Darling test–Martinez-Iglewicz test–Kolmogorov-Smirnov test–D’Agostino Omnibus test
Shapiro-Wilk W test
• Q-Q Plot method
Correlation Statistics
• Correlation• Relation between two variables
Types of Correlations
Correlation Coefficient
• Pearson’s Correlation Coefficient ( r)• Spearman’s Rank Correlation Coefficinet (R )
• Perfect Positive Correlation = +1• Perfect Negative Correlation = -1• No Correlation = 0
Interpretation
Size of Correlation Interpretation.90 to 1.00 (−.90 to −1.00) Very high positive (negative)
correlation.70 to .90 (−.70 to −.90) High positive (negative)
correlation.50 to .70 (−.50 to −.70) Moderate positive (negative)
correlation.30 to .50 (−.30 to −.50) Low positive (negative)
correlation.00 to .30 (.00 to −.30) negligible correlation
Example – Multiple Correlation
SL NO GENDER DIET SKIP MEALS STRESS SLEEP PRAKRITI KOSHTA SATWA PARINAMASO
OLA SCORE
1 1 1 1 1 1 2 2 2 6
2 2 2 1 1 1 2 2 2 6
3 1 2 1 1 1 2 2 2 5
4 1 1 0 0 0 1 1 3 2
5 2 2 0 0 1 2 2 2 6
6 2 1 1 1 1 2 2 2 6
7 2 2 1 1 1 2 2 2 6
8 1 1 1 0 0 3 1 3 3
9 2 1 1 1 1 2 2 2 6
10 1 1 0 0 0 3 3 1 2
Multiple Correlation Matrix
GENDER DIETSKIP
MEALS STRESS SLEEP PRAKRITI KOSHTA SATWA
PARINAMASOOLA SCORE
GENDER 1
DIET 0.408248 1
SKIP MEALS 0.218218 0.089087 1
STRESS 0.408248 0.25 0.801784 1
SLEEP 0.654654 0.534522 0.52381 0.801784 1
PRAKRITI -0.1857 -0.15162 0.121566 -0.22743 -0.28365 1
KOSHTA 0.185695 0.15162 -0.12157 0.227429 0.283654 0.37931 1
SATWA -0.1857 -0.15162 0.121566 -0.22743 -0.28365 -0.37931 -1 1
PARINAMASOOLA SCORE 0.722315 0.4669 0.577948 0.761783 0.972003 -0.2012 0.201196 -0.2012 1
Regression
• Regression = PredictionH
EIG
HT
WEIGHT
REGRESSION EQUATION
y = a + bx
a = Interceptb = Slope
Example - Regression
SMOKING PARINAMASOOLA SCORE
1 61 61 50 21 61 61 60 31 61 5
CORRELATION COEFFICIENT
= 0.945
0 50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.085
0.15
0.225
0.28
0.425
0.535f(x) = 0.0026 x + 0.0233333333333334
Regression Statistics
Multiple R 0.9456109
R Square 0.8941799
Adjusted R Square 0.8809524
Standard Error 0.5
Observations 10
ANOVA
df SS MS F Significance F
Regression 1 16.9 16.9 67.6 0.0001
Residual 8 2 0.25
Total 9 18.9
Example – Multiple Regression
SL NO GENDER DIET SKIP MEALS STRESS SLEEP PRAKRITI KOSHTA SATWA PARINAMASO
OLA SCORE
1 1 1 1 1 1 2 2 2 6
2 2 2 1 1 1 2 2 2 6
3 1 2 1 1 1 2 2 2 5
4 1 1 0 0 0 1 1 3 2
5 2 2 0 0 1 2 2 2 6
6 2 1 1 1 1 2 2 2 6
7 2 2 1 1 1 2 2 2 6
8 1 1 1 0 0 3 1 3 3
9 2 1 1 1 1 2 2 2 6
10 1 1 0 0 0 3 3 1 2
Regression StatisticsMultiple R 0.99R Square 0.98Adjusted R Square 0.28Standard Error 0.33Observations 10
Coefficien
tsStandard Error t Stat P-value
Intercept 0.333 0.660 0.505 0.648
GENDER 0.500 0.289 1.732 0.182
DIET -0.333 0.272 -1.225 0.308
STRESS -0.167 0.397 -0.420 0.703
SLEEP 2.333 0.569 4.099 0.026
PRAKRITI 0.500 0.236 2.121 0.124
KOSHTA 1.000 0.236 4.243 0.024
Statistical Software
• SPSS• SAS• STAT• R• INSTAT• Epi Info
Thank you
