Hypothesis testing lectures

HYPOTHESIS TESTING
PRESENTED BY:
DR SANJAYA KUMAR SAHOO
PGT,AIIH&PH,KOLKATA

HYPOTHESIS TESTING:
OBJECTIVE:
 To test whether evidence for assumption or statements we make
about our research objectives(i.e alternative hypothesis) against the
previous or existing history of that particular research objective(i.e
null hypothesis)

What is hypothesis?
• Hypothesis is a belief concerning a parameter (i.e statistical constant)
such as prevalence,incidence,population mean,correlation etc.
• Example:
 incidence of TB is higher among low SES groups as compared to
higher SES groups.
Incidece of lung cancer is higher among smokers as compared to non-
smokers.

TESTING OF HYPOTHESES:
• states a hypothesis to be tested,
• formulates an analysis plan,
• analyzes sample data according to
the plan, and
• accepts or rejects the hypothesis,
based on results of the analysis.
The
researcher
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The scientific hypothesis, is of 2 types:
1.null hypothesis (denoted by H0)
2.alternative hypothesis (denoted by H1).
Thus the alternative hypothesis is the assertion
accepted when null hypothesis is rejected

NULL HYPOTHESIS (H0):
 Null hypothesis is generally a statement that assumes that, there is “NO”
difference between two sets of values.
Examples:
• There is no difference between 2 drugs A & B.
• There is no association between lung cancer and smoking.
• Mean cholesterol level in normals =mean cholesterol level in HTN patients.

ALTERNATIVE HYPOTHESIS(H1)
• It states that there is difference between two sets of values.
Examples:
• There is difference between 2 drugs A & B.
• There is an association between lung cancer and smoking.
• Mean cholesterol values in normals ≠ mean cholesterol in HTN patients.

TYPES OF ERRORS:
TYPE 1 ERROR:
 When null hypothesis H0 is true but still it is rejected type 1 error(alpha error) is
committed .
It is represented as α(alpha) or p-value or level of significance.
Usually P is less than 5 in a hundred(p<0.05)
TYPE II ERROR:
 When null hypothesis H0 is false but it is accepted/fails to reject type-2 error
(beta error) is committed.
 Power of test (1-β) is the ability of test to correctly reject H0(i.e no diff between
the drugs) in favour of H1(i.e there is a difference) when H0 is false.

H0 is true
H0 rejected
H0 is false
H0 accepted

H0 -> μAtorva(A)= μRosuva(B)
Ha->μAtorva(A)≠ μRosuva(B) (2 tailed)
Ha->μAtorva(A)> μRosuva(B) (1 tailed)
Ha->μAtorva(A)< μRosuva(B) (1 tailed)

The power (1-β) of a study
The probability that it correctly
 rejects the null hypothesis(A=B) when the null hypothesis is false.
 accepts the alternative hypothesis(A≠B) when the alternative hypothesis
is true –(It assumes that there is a difference)
 identifies a significant difference or effect or association in the sample
that exists in the population.
The larger the sample size, the study will have greater power
• “Power” is the antithesis of “risk of Type II error”
• Risk of Type II error = 1 – power
• Power = 1 - Risk of Type II error

Type(Alpha) Error Z=standardized
normal deviate
P Z(alpha)
0.05 1.96
0.01 2.57
0.001 3.29
Type(Beta) Error
p Z(beta)
0.10 1.282
0.15 1.037
0.20 0.842
0.25 .675

Hypothesis testing
 When mean is the parameter of the study and
 σ (sigma)→ pooled standard deviation/common
sd between 2 groups
 d → clinically meaningful difference between
two groups/difference between 2 means
 Zα → Z value for α level (type I error)
 Zβ → Z value for β level (type II error)

Estimation of differences between two
means (on quantitative variable like BP,
blood sugar, serum cholesterol etc.)
Requirements:
Estimate of variables of individual values (means &
standard deviations)
Magnitude of differences that is desired to detect(limit
of accuracy required)
Degree of confidence required
The value of the power desired (1-)
 Larger the difference, smaller the trial
size.
 Greater the Power, greater the trial
size.
 Larger the s.d., greater the trial size.

Simple formula When mean is the parameter of
the study :
Sample size in each
group (assumes equal
sized groups)
Represents the
desired power
(typically .84 for 80%
power).
Represents the desired
level of statistical
significance (typically
1.96).
Common Standard deviation between
two groups
σ2 =(n1-1)σ1
2 +(n2-1)σ2
2/(n1+n2)-2
Effect Size (the
difference in
means)
2
2
/2
2
difference
)Z(2  

Z
n

Hypothesis testing:
 When proportion is the parameter of the study and
 p1 → proportion in the first group
 p2 → proportion in the second group
P
 d (p1 - p2 ) → clinically meaningful difference between proportion
of two groups
 Zα → Z value for α level (type I error)
 Zβ → Z value for β level (type II error)
Average of P1 & P2

Estimation of difference between two
proportions
Requirements:
• An estimate of response rate in two groups
• Difference in response rates
• Level of statistical significance ()
• The value of the power desired (1-)
• Whether the test should be one sided or two sided
1.Larger the difference, smaller the trial
size
2. Larger the Power, larger the trial size
3. Absolute value of average P, also
affects trial size

Simple formula when proportion is the
parameter of the study
2
21
2
/2
)(p
)Z)(1)((2
p
Zpp
n


 
Sample size in each
group (assumes equal
sized groups)
Represents the
desired power
(typically .84 for 80%
power).
Represents the
desired level of
statistical
significance
(typically 1.96).
A measure of
variability (similar to
standard deviation)
Effect Size (the
difference in
proportions)

EFFECT SIZE
 Crucial factor : comparative studies
 Smallest measured difference between the comparison
groups
 Unknown, selection of a reasonable difference derived
from pilot studies/lit review/past experience
 ES or the clinically meaningful difference is determined
by the investigator, not by statistician
 For small ES sample size is large
 The investigator may choose a minimum expected
difference of 10%(.1) and this is effect size.
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Hypothesis testing lectures

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