How to Learn Everything You Ever Wanted to Know About Statistics

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Two-samples tests, X2
Hypothesis Testing
Dr. Mona Hassan Ahmed Prof. of Biostatistics HIPH,
Alexandria University
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Z-test (two independent proportions)

• P1 proportion in the first group
• P2 proportion in the second group
• n1 first sample size
• n2 second sample size

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• Critical z
• 1.96 at 5 level of significance
• 2.58 at 1 level of significance

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Example

• Researchers wished to know if urban and rural
  adult residents of a developing country differ
  with respect to prevalence of a certain eye
  disease. A survey revealed the following
  information

Total Eye disease Eye disease Residence
Total No Yes Residence
300 276 24 Rural
500 485 15 Urban
Test at 5 level of significance, the difference
in the prevalence of eye disease in the 2 groups
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Answer

• P1 24/300 0.08 p2 15/500 0.03

2.87 gt Z The difference is statistically
significant
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t-Test (two independent means)
mean of the first group
mean of the second group
S2p pooled variance
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• Critical t from table is detected
• at degree of freedom n1 n2 - 2
• level of significance 1 or 5

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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

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Answer
S1 12 S2 11
State H0 H0 ?1 ?2 State H1
H1 ?1 ? ?2 Choose a a 0.01
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Answer

• Critical t at df 40 1 level of significance
  2.58

Decision 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 .
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Degrees of freedom Probability (p value) Probability (p value) Probability (p value)
Degrees of freedom 0.10 0.05 0.01
1 6.314 12.706 63.657
5 2.015 2.571 4.032
10 1.813 2.228 3.169
17 1.740 2.110 2.898
20 1.725 2.086 2.845
24 1.711 2.064 2.797
25 1.708 2.060 2.787
? 1.645 1.960 2.576
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Paired t- test (t- difference)

• Uses
• To compare the means of two paired samples.
• Example, mean SBP before and after intake of drug.

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• di difference (after-before)
• Sd standard deviation of difference
• n sample size
• Critical t from table at df n-1

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Example

• The following data represents the reading of SBP
  before and after administration of certain drug.
  Test whether the drug has an effect on SBP at 1
  level of significance.

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SBP (After) SBP (Before) Serial No.
180 200 1
165 160 2
175 190 3
185 185 4
170 210 5
160 175 6
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Answer
di2 di After-Before BP After BP Before Serial No.
400 -20 180 200 1
25 5 165 160 2
225 -15 175 190 3
0 0 185 185 4
1600 -40 170 210 5
225 -15 160 175 6
2475 -85 Total
? di2 ?di Total
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Answer
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Answer

• Critical t at df 6-1 5 and 1 level of
  significance
• 4.032
• Decision
• Since t is lt critical t so there is no
  significant difference between mean SBP before
  and after administration of drug at 1 Level.

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Degrees of freedom Probability (p value) Probability (p value) Probability (p value)
Degrees of freedom 0.10 0.05 0.01
1 6.314 12.706 63.657
5 2.015 2.571 4.032
10 1.813 2.228 3.169
17 1.740 2.110 2.898
20 1.725 2.086 2.845
24 1.711 2.064 2.797
25 1.708 2.060 2.787
? 1.645 1.960 2.576
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Chi-Square test

• It tests the association between variables... The
  data is qualitative .
• It is performed mainly on frequencies.
• It determines whether the observed frequencies
  differ significantly from expected frequencies.

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Where E expected frequency O
observed frequency
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• Critical X2 at df (R-1) ( C -1) Where R raw
  C column
• I f 2 x 2 table
• X2 3.84 at 5 level of significance
• X2 6.63 at 1 level of significance

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Example

• In a study to determine the effect of heredity in
  a certain disease, a sample of cases and controls
  was taken

Total Disease Disease Family history
Total Control Cases Family history
200 120 80 Positive
300 160 140 Negative
500 280 220 Total
Using 5 level of significance, test whether
family history has an effect on disease
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Answer
Total Disease Disease Family history
Total Control Cases Family history
200 120 112 80 88 positive O E
300 160 168 140 132 Negative O E
500 280 220 Total

• X2 (80-88)2/88 (120-112)2/112
  (140-132)2/132 (160-168)2/168
• 2.165 lt 3.84
• Association between the disease and family
  history is not significant

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Odds Ratio (OR)

• The odds ratio was developed to quantify exposure
  disease relations using case-control data
• Once you have selected cases and controls ?
  ascertain exposure
• Then, cross-tabulate data to form a 2-by-2 table
  of counts

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2-by-2 Crosstab Notation
Disease Disease - Total
Exposed A B AB
Exposed - C D CD
Total AC BD ABCD

• Disease status AC no. of cases
• BD no. of non-cases
• Exposure status AB no. of exposed individuals
• CD no. of
  non-exposed individuals

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The Odds Ratio (OR)
Disease Disease -
Exposed A B
Exposed - C D

Cross-product ratio
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Example

• Exposure variable Smoking
• Disease variable Hypertension

D D-
E 30 71
E- 1 22
Total 31 93
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Interpretation of the Odds Ratio

• Odds ratios are relative risk estimates
• Relative risk are risk multipliers
• The odds ratio of 9.3 implies 9.3 risk with
  exposure

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Interpretation
Positive association Higher risk
OR gt 1
No association
OR 1
OR lt 1
Negative association Lower risk (Protective)
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OR Confidence level

• In the previous example
• OR 9.3
• 95 CI is 1.20 72.14

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Multiple Levels of Exposure
Smoking level Cases Controls
Heavy smokers 213 274
Moderate smokers 61 147
Light smokers 14 82
Non-smokers 8 115
Total 296 618
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Multiple Levels of Exposure

• k levels of exposure ? break up data into (k 1)
  2?2 tables
• Compare each exposure level to non-exposed
• e.g., heavy smokers vs. non-smokers

Cases Controls Controls
Heavy smokers 213 213 274
Non-smokers 8 8 115
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Multiple Levels of Exposure
Smoking level Cases Controls
Heavy smokers 213 274 OR3 (213)(115)/(274)(8)11.2
Moderate smokers 61 147 OR2 (61)(115)/(147)(8) 6.0
Light smokers 14 82 OR1 (14)(115)/(82)(8) 2.5
Non-smokers 8 115
Total 605 115
Notice the trend in OR (dose-response
relationship)
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Small Sample Size Formula For the Odds Ratio

• It is recommend to add ½ to each cell before
  calculating the odds ratio when some cells are
  zeros

D D-
E 31 71
E- 0 22
Total 31 93
OR Small Sample (A0.5)(D0.5)
OR Small Sample (B0.5)(C0.5)
OR Small Sample (310.5)(220.5) 19.8
OR Small Sample (710.5)(00.5) 19.8
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