SPEAKER:-Shubhanshu gupta

TEACHER I/C:-Dr.S.K.Verma

DATE:-16/09/2014

1

Historical aspect.

Basis of statistical inference.

Hypothesis and it’s testing.

Characteristics of Hypothesis.

Null Hypothesis.

Alternate Hypothesis.

Interpreting the result of Hypothesis.

Type I error and Type II error.

One-tailed test , Two-tailed test.

Effect of sample size.

Test of significance.

Parametric Vs Non Parametric test.

Parametric test.

Non-Parametric test.

References. 2

The term statistical significance was coined by Ronald Fisher(1890-1962).

Student t-test : William Sealy Gosset.

3

Statistical inference is the branch of

statistics which is concerned with using

probability concept to deal with

uncertainly in decision making.

It refers to the process of selecting and

using a sample to draw inference about

population from which sample is drawn.

4

Statistical Inference

Estimation of

population value

Testing of

hypothesis

Point

estimation

Range

estimation

Mean,

proportion

estimation

Confidence

interval

estimation

5

• During investigation there is assumption and

presumption which subsequently in study must be

proved or disproved.

• Hypothesis is a supposition made from observation. On

the basis of Hypothesis we collect data.

• Hypothesis is a tentative justification, the validity of

which remains to be tested.

Two Hypothesis are made to draw inference from

Sample value-

A. Null Hypothesis or hypothesis of no difference.

B. Alternative Hypothesis of significant difference.6

The Null Hypothesis is symbolized as Ho and

Alternative Hypothesis is symbolized as H1 or HA.

In Hypothesis testing we proceed on the basis of

Null Hypothesis. We always keep Alternative

Hypothesis in mind.

The Null Hypothesis and the Alternative

Hypothesis are chosen before the sample is drawn.

7

1. Hypothesis should be clear and precise.

2. Hypothesis should be capable of being tested.

3. It should state relationship between variables.

4. It must be specific.

5. It should be stated as simple as possible.

6. It should be amenable to testing within a

reasonable time.

7. It should be consistent with known facts.

8

A Null Hypothesis or Hypothesis of no difference

(Ho) between statistic of a sample and parameter of

population or between statistic of two samples

nullifies the claim that the experimental result is

different from or better than the one observed

already. In other words, Null Hypothesis states that

the observed difference is entirely due to sampling

error, that is - it has occurred purely by chance.

9

There is no difference between the operational procedures of open prostatectomy and TURP.

There is no difference between open operation and transsphenoidal approach.

There is no difference in the incidence of measles between vaccinated and non-vaccinated children.

Drugs chloramphenicol is as good as drug cotrimoxazole in treating enteric fever.

10

Alternative Hypothesis of significant difference

states that the sample result is different that is,

greater or smaller than the hypothetical value of

population.

A test of significance such as Z-test, t-test, chi-

square test, is performed to accept the Null

Hypothesis or to reject it and accept the Alternative

Hypothesis.

11

The Hypothesis Ho is true - our test accepts it because the result falls within the zone of acceptance at 5% level of significance.

The Hypothesis Ho is false - test rejects it because the estimate falls in the area of rejection.

12

Zone of acceptance- If the results of a sample falls

in the plain area i.e. within the mean+/-1.96

standard error the Null Hypothesis is accepted- the

area is called zone of acceptance.

Zone of rejection-If the result of a sample falls

outside the plain area, i.e. beyond mean +/-1.96

standard error, it is significantly different from

population value. So Null Hypothesis is rejected and

alternative hypothesis is accepted. This area is

called zone of rejection.

13

Accept H0

Reject H0 Reject H0

Zcrit Zcrit

Setting a criterion

0H

Null Hypothesis

14

When a Null Hypothesis is tested, there may be four

possible outcomes:

i. The Null Hypothesis is true but our test rejects it.

ii. The Null Hypothesis is false but our test accepts it.

iii. The Null Hypothesis is true and our test accepts it.

iv. The Null Hypothesis is false but our test rejects it.

Type 1 Error – rejecting Null Hypothesis when Null

Hypothesis is true. It is called ‘α error’.

Type 2 Error – accepting Null Hypothesis when Null

Hypothesis is false. It is called ‘β-error’.

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Decision

Reject Ho Accept Ho

Type 1 errorCorrect decision

Ho true

Correct decision

Type 2 error Ho false

16

Decision

More effective Same effect

Error Correct decision

New regime is not better

Correct decisionErrorNew regime is better

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The Null Hypothesis is

False True

β (type 2 error)1-α (confidence level)

Accept if p>=0.05(non-significant) conclusion-negative

1-β (power of the test)

α (type 1 error)Reject if p<0.05(significant) conclusion-positive

18

The probability of committing Type 1 Error is called the

P-value. Thus p-value is the chance that the presence

of difference is concluded when actually there is none.

When the p value is between 0.05 and 0.01 the result is

usually called significant.

When p value is less than 0.01, result is often called

highly significant.

When p value is less than 0.001 and 0.005, result is

taken as very highly significant.

19

The statistical power of a test is the

probability that a study or a trial will be

able to detect a specified difference . This

is calculated as 1- probability of type II

error, i. e. probability of correctly

concluding that a difference exists when it

is indeed present. Thus, power = 1-β.

20

Confidence Interval : The interval within which

a parameter value is expected to lie with a

certain confidence level as could be revealed

by repeated samples is called confidence

interval.

Confidence Level : The degree of assurance for

an interval to contain the value of a parameter

(1-α).

21

If HA states is < some value, critical region

occupies left tail.

If HA states is > some value, critical region

occupies right tail.

ONE-TAILED TEST

22

H0: µ = 100

H1: µ > 100

Values that differ “significantly”

from 100100

Points Right

Fail to reject H0 Reject H0

RIGHT-TAILED TEST

alpha

Zcrit

23

H0: µ = 100

H1: µ < 100

100

Values that differ “significantly”

from 100

Points Left

Fail to reject H0Reject H0

LEFT-TAILED TESTS

alpha

Zcrit

24

• HA is that µ is either greater or less than µH0

HA: µ ≠ µH0

• is divided equally between the two tails of

the critical region.

TWO-TAILED HYPOTHESIS TESTING

25

H0: µ = 100

H1: µ 100

100

Values that differ significantly from 100

Means less than or greater than

Fail to reject H0Reject H0 Reject H0

alpha

Zcrit Zcrit

TWO-TAILED HYPOTHESIS

TESTING

26

With large n (say, n > 30), assumption of normal population

distribution not important.

For a given observed sample mean and standard deviation, the

larger the sample size n, the larger the test statistic (because

denominator is smaller) and the smaller the P-value.

We’re more likely to reject a false H0 when we have a larger

sample size (the test then has more “power”)

With large n, “statistical significance” not the same as “practical

significance.”

27

Test of significance is a formal procedure for comparing observed data with a claim (also called a hypothesis) whose truth we want to assess.

Test of significance is used to test a claim about an unknown population parameter.

A significance test uses data to evaluate a hypothesis by comparing sample point estimates of parameters to values predicted by the hypothesis.

We answer a question such as, “If the hypothesis were true, would it be unlikely to get data such as we obtained?”

28

Based on specific

distribution such

as Gaussian

Not based on any

particular parameter such

as mean

Do not require that the

means follow a particular

distribution such as

Gaussian.

Used when the underlying

distribution is far from

Gaussian (applicable to

almost all levels of

distribution) and when the

sample size is small

29

Parametric Tests

Student’s t- test(one

sample, two sample,

and paired)

Z test

ANOVA F-test

Pearson’s correlation(r)

Non-Parametric

TestsSign test(for paired data)

Wilcoxon Signed-Rank

test for matched pair

Wilcoxon Rank Sum test

(for unpaired data)

Chi-square test

Spearman’s Rank

Correlation(p)

ANOCOVA

Kruskal-Wallis test

30

Purpose of

application

Parametric test Non-Parametric test

Comparison of two

independent groups.

‘t’-test for independent

samples

Wilcoxon rank sum test

Test the difference

between paired

observation

‘t’-test for paired

observation

Wilcoxon signed-rank

test

Comparison of several

groups

ANOVA Kruskal-Wallis test

Quantify linear

relationship between

two variables

Pearson’s Correlation Spearman’s Rank

Correlation

Test the association

between two qualitative

variables

_ Chi-square test

31

Students t- tests - A statistical criterion to test the hypothesis that mean is superficial value, or that specified difference, or no difference exists between two means. It requires Gaussian distribution of the values, but is used when SD is not known.

Proportion test - A statistical test of hypothesis based on Gaussian distribution, generally used to compare two means or two proportions in large samples, particularly when the SD is known.

ANOVA F-test - used when the number of groups compared are three or more and when the objective is to compare the means of a quantitative variable.

32

One sample– only one group is studied and an

externally determined claim is examined.

Two sample– there are two groups to

compare.

Paired– used when two sets of measurements

are available, but they are paired .

33

Find the difference between the actually observed

mean and the claimed mean.

Estimate the standard error (SE) of mean by S/n,

where s is the standard deviation and n is the

number of subjects in the actually studied sample.

The SE measures the inter-sample variability

Check the difference obtained in step 1 is

sufficiently large relative to the SE. for this ,

calculate students t. this is called the test

criterion. Rejection or non-rejection of the null

depends on the value of this t .

Reject the null hypothesis if the t-value so

calculated is more than the critical value

corresponding to the pre-fixed alpha level of

significance and appropriate df.

34

There are 10 patients of arthritis. Suppose the reduction in pain

after using newspirin is as follows on a 10-point visual analog

scale:

Mean reduction, x = 2.3 points and SD, s=2.11.

By using the formula :

t = 2.3-3.0/(2.11/10^½)

= -1.049

n = 10, df = 10-1 = 9

For one-tailed α = 0.05, and df = 9 , the critical value of t is 1.833

Since the calculated value 1.049 of t is less than the critical value

1.833, the Null Hypothesis that the mean reduction in pain is 3

point can not be rejected.

0 3 6 1 1 4 0 2 1 5

35

Sp – is the pooled SD.

36

A study on 24-hour creatinine excretion in male

and female healthy adults to examine if a

difference exists. For our illustration ,we give

value obtained for 15 subjects in group in table:Me

n

16

.6

19

.8

17

.1

15

.6

20

.3

24

.7

18

.5

17

.6

22

.0

24

.9

18

.4

16

.9

21

.1

17

.0

23

.3

W

o

m

en

23

.2

22

.0

21

.9

14

.2

23

.2

24

.8

25

.5

28

.1

21

.8

20

.9

18

.0

19

.5

20

.6

16

.7

17

.3

37

df = n1+n2-2 = 15+15-2 = 28

in men, y1 = 19.59 and s1 = 3.03

in women, y2 = 21.18 and s2 = 3.65

sp = [(15-1)x(3.03)^²+(15-1)x(3.65) ^²/15+15-2]^½

=3.35

Thus,

t = 19.59-21.18/3.35(1/15+1/15) ^½

= -1.59/1.2232 = -1.30

The critical value of t is 2.048, the calculated value is less than the critical value. Thus the Null Hypothesis of equality can not be rejected.

38

Obtain the difference for each pair and test

the null hypothesis that the mean of these

differences is zero(this null hypothesis is

same as saying that the means before and

after are equal).

For paired samples : t = d/(Sd/(n)^1/2)

d : is the sample mean of the

differences

Sd : is standard deviation.

39

Consider serum albumin level of 8 randomly

chosen patients of dengue haemorrhagic fever

before and after treatment. The value has been

tabulated :

Before

treatm

ent

5.1 3.8 4.0 4.7 4.5 4.8 4.1 3.6

After

treat

ment

4.8 3.7 3.8 4.7 4.6 5.0 4.0 3.4

Differ

ence(

d)

0.3 0.1 0.2 0 -0.1 -0.2 0.1 0.2

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Mean difference, d = 0.6/8 = 0.075g/dl, and SD of

difference, sd = 0.17.

t = 0.075/0.17/(8)^½ = 1.25

df = 8-1 =7

The critical value of t is 2.365, since the calculated

value is less, the null hypothesis of difference can not

be rejected.

41

Used for large Quantitative data (i.e. n>30) .

Application: To find out Standard Error of difference between two sample means

i.e. S. E. (X1 - X2)

e.g. To find our significant differencebetween two different variables/groups i.e.

Efficacy of two drugs, difference between two

groups etc.

42

State the Null Hypothesis i.e. H0 and its

Alternative Hypothesis i.e. H1

Find out the values of test statistic i.e. value

of 'Z' as follows:

_ _ _ _

Z = X1 – X2 / SE (X1 – X2)

where,

SE (X1 – X2)=√ (SD1)²/n1 + (SD2)² /n2

43

Situations where it is used are1.in two sample situation2. in paired set-up3.in repeated measures, when the same subject is measured at different time points such as after 5 minutes, 15 minutes, 30 minutes, 60 minutes etc,.4.removing the effect of a covariate5. regression.

44

Correlation is the relationship between two or more paired factors or two or more sets. The degree of relationship is usually measured and represented by a correlation coefficient.

A correlation coefficient is numerical measure of the linear relationship between two factors or sets of scores. Coefficient can be identified by either the letter r or the Greek letter rho. Or other symbols, depending on the manner the coefficient has been computed. Obtained correlation

45

The sign of the obtained correlation coefficient

can range from coefficient indicates the directions

of the relationship and the numerical value of its

strength.

Correlation Coefficient Degree of Relationship

.00 - .20 Negligible

.21 - .40 Low

.41 - .60 Moderate

.61 - .80 Substantial

.81 – 1.00 High to Very High

46

Types of Correlation :

Types

Type 1 Type 2 Type 3

Type1

Positive Negative No Perfect47

Type 2

Linear Non – linear

Type 3

Simple Multiple Partial

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r = [NSXY- (SX) (SY)] /[(NSX² - (SX²) - (NSY² - (SY) ²)]^½

Where:

N = Number of paired observation

SXY = sum of the cross products of C and Y

SX = sum of the scores under Variable X

SY = sum of the scores under variable Y

(SX)² = Sum of x scores acquired

(SY) = sum of y equated

SX² = sum of squared X scores

SY² = Sum of squared Y scores

49

Alternative to the test of significance of

difference between two proportions

O : Observed frequencies.

E : Expected frequencies.

50

Do you know that prevalence of cataract is more in

males or in females? Consider a study on prevalence of

cataract in males and females of age 50 years and

above. The results are as follows. Number of males

examined (n1) = 60: found with cataract 37. Number of

females examined (n2) = 40 : found with cataract 30.

This is stated in a table format

Gender Yes No Total

Male 37 23 60

Female 30 10 40

Total 67 33 10051

Expected frequency = Corresponding row total X Corresponding

column total / Grand total

=60x33/100 = 19.8

Applying the formula

=(37-40.2)^2/40.2+(23-19.8) ^2/19.8+(30-26.8) ^2/26.8+(10-

13.2) ^2/13.2

= 0.2547+0.5172+0.3821+0.7758

= 1.93

The critical value of chi-square is 3.84 at 5% level of significance.

Since the calculated value is less than the critical value, the Null

Hypothesis can not be rejected.

52

For paired data

It is a non parametric test based on

signs(positive and negative) of the

differences in the levels seen before and

after therapy .

53

For matched pairs.

It is better test than the sign test– assigns rank to the differences of n pairs after ignoring the + or – signs.

The lowest difference gets rank 1 and the highest gets rank n.

Sum of the only those ranks that are associated with positive difference obtained(Wilcoxon signed rank criteria).

It is similar to Mann-Whitney test.

54

For unpaired two sample situation .

If there are n1 subjects in the first sample

and n2 in the second sample, these(n1+n2)

values are jointly ranked from 1 to (n1+n2)

{the sum of these ranks is obtained for those

subjects only who are in smaller group}.

55

Spearman’s correlation is designed to measure the

relationship between variables measured on an ordinal

scale of measurement.

Similar to Pearson’s Correlation, however it uses ranks

as opposed to actual values.

56

1. Convert the observed values to ranks (accounting for

ties)

2. Find the difference between the ranks, square them and

sum the squared differences.

3. Set up Hypothesis, carry out test and conclude based on

findings.

4. If the Null is rejected then calculate the Spearman

correlation coefficient to measure the strength of the

relationship between the variables.

57

Where, di is the difference between the paired ranksn is the number of pairs.

The Spearman rank correlation coefficient may lie between -1 to +1. Values close to +/-1 indicate a high correlation ; values close to zero indicate lack of relationship.

)1(

6

12

1

2

NN

n

iid

58

A Indrayan and L Satyanarayana-biostatistics, 20006 ed, Printice -Hall of India.

MSN Rao, NS Murthy-applied statistics in health sciences, 2nd ed, 2010, jaypee.

B Antonisamy, Solomon Christopher, P Prasanna Samuel – Boistatistics Principles and Practice.

www. Wikipedia. org

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