Test of-significance : Z test , Chi square test

Tests of significance

Qualitative data

Z & Chi square Test

Dr. Shaikh B.M.

JRII

Dept. of Community Medicine

outline

Basic terms (recap)……….

Sampling Variation,

Null hypothesis,

Level of significance and confidence,

P value, power of the test, degree of freedom

Tests of significance and type

Selection of the test of significance

Steps in hypothesis testing

Z test (SEDP)

Chi Square test

Limitations of the tests of significance

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Sampling Variation

• Research done on samples and not on populations.

• Variability of observations occur among different samples.

• This complicates whether the observed difference is due to

biological or sampling variation from true variation.

• To conclude actual difference, we use tests of significance

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Test of significance

• The test which is done for testing the research hypothesis

against the null hypothesis.

Why it is done?

To assist administrations and clinicians in making decision.

The difference is real ? or

Has it happened by chance ?

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Null Hypothesis (HO)

• 1st step in testing any hypothesis.

• Set up such that it conveys a meaning that there exists no

difference between the different samples.

• Eg: Null Hypothesis – The mean pulse rate among the two

groups are same (or) there is no significant difference between

their pulse rates.

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• By using various tests of significance we either:

–Reject the Null Hypothesis

(or)

–Accept the Null Hypothesis

• Rejecting null hypothesis → difference is significant.

• Accepting null hypothesis → difference is not significant.

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Level of significance and confidence

• Significance means the percentage risk to reject a null

hypothesis when it is true and it is denoted by 𝛼.Generally

takenas1%,5%,10%

• (1 − 𝛼) is the confidence level in which the null hypothesis will

exist when it is true.

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Level of Significance – “P” Value

• p-value is a function of the observed sample results

(a statistic) that is used for testing a statistical hypothesis.

• It is the probability of null hypothesis being true. It can accept

or reject the null hypothesis based on P value.

• Practically, P < 0.05 (5%) is considered significant.

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• P = 0.05 implies,

– We may go wrong 5 out of 100 times by rejecting null

hypothesis.

– Or, We can attribute significance with 95% confidence.

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5% Significance level & 95% confidence level

Acceptance and Rejection

regions

𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑔𝑖𝑜𝑛/𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙

(𝛼 = 0.025 𝑜𝑟 2.5%)

𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑔𝑖𝑜𝑛/𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙

(𝛼 = 0.025 𝑜𝑟 2.5%)

𝑇𝑜𝑡𝑎𝑙 𝐴𝑐𝑐𝑒𝑝𝑡𝑎𝑛𝑐𝑒 𝑟𝑒𝑔𝑖𝑜𝑛𝑜𝑟

𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙

(1 − 𝛼) = 95%

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Power of the test &

Degree of freedom

Power of the test :

Type II error is β and 1−β is called power of the test.

Probability of rejecting False H0, i. e. taking correct decision

Degree of freedom:

• Number of independent observations used in statistics (d. f.)

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Various tests of significance

1. Parametric – Data is or assumed to be normal distributed.

2. Non parametric – Data is not normal distributed.

Parametric tests :

For Qualitative data:-

1. Z test

2. 2. Chi-square test or X2

For Quantitative data:-

1. Unpaired ‘t’ test

2. Paired ‘t’ test

3. ANOVA12

Selection of the test

• Base

1. Type of data

2. Size of sample

3. Number of samples

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Selection of the test

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Qualitative

Data

n > 30

Large Sample

1, 2 samples

Z test

n </> 30

Small / large Sample

More than 2 samples

Chi square test

Quantitative

Data

n </> 30

Small / large Sample

1 samples

One sample

t-test

2 samples

Unpaired t-test / Paired t-

test

More than 2 samples

ANOVA

Steps in Testing a Hypothesis

• General procedure in testing a hypothesis

1. Set up a null hypothesis (HO).

2. Define alternative hypothesis (HA).

3. Calculate the test statistic (Z, X2, t etc.).

4. Find out the corresponding probability level (P Value) for the calculated test statistic from relevant tables.

5. Accept or reject the Null hypothesis depending on P value.

P> 0.05 HO accepted

P < 0.05 HO rejected

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Z test[Standard error of difference between proportions (SEp1-p2)]

• For comparing qualitative data between 2 groups.

• Used for large samples only. (> 30 in each group).

• Standard error of difference between proportions (SEp1-p2) is

calculated by using the formula

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• And then the test statistic ‘Z Score’ value is calculated by

using the formula:

• If Z > 1.96 P < 0.05 SIGNIFICANT - H0 rejected

• If Z < 1.96 P > 0.05 NOT SIGNIFICANT- H0 accepted

• Finding p value from Z table

• Go to the row that represents the ones digit and the first digit

after the decimal point (the tenths digit) of your z-value.

• Go to the column that represents the second digit after the

decimal point (the hundredths digit) of your z-value.

• Intersect the row and column from Steps 1 and 2.

1.0000 - 0.9772 =0.0228< 0.05

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Example

• Consider a hypothetical study where cure rate of Typhoid fever

after treatment with Ciprofloxacin and Ceftriaxone were

recorded to be 90% and 80% among 100 patients treated with

each of the drug.

• How can we determine whether cure rate of Ciprofloxacin is

better than Ceftriaxone?

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Solution

Step – 1: Set up a null hypothesis – H0:

– “There is no significant difference between cure rates of

Ciprofloxacin and Ceftriaxone.”

Step – 2: Define alternative hypothesis – Ha:

– “Ciprofloxacin is 1.125 times better in curing typhoid fever

than Ceftriaxone.”

Step – 3: Calculate the test statistic – ‘Z Score’

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Step – 3: Calculating Z score

– Here, P1 = 90, P2 = 80

– SEP1-P2 will be given by the formula:

So, Z = 10/5 = 2

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Step – 4: Find out the corresponding P Value

– Since Z = 2 i.e., > 1.96, hence, P < 0.05

Step – 5: Accept or reject the Null hypothesis

– Since P < 0.05, So we reject the null hypothesis(H0)

There is no significant difference between cure rates of

Ciprofloxacin and Ceftriaxone

– And we accept the alternate hypothesis (Ha)

that, Ciprofloxacin is 1.125 times better in curing typhoid

fever than Ceftriaxone

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Chi square (X2) test

• This test is also for testing qualitative data.

• Its advantage over Z test is:

– Can be applied for smaller samples as well as for large

samples.

• Prerequisites for Chi square (X2) test to be applied:

– The sample must be a random sample

– None of the observed values must be zero.

– Adequate cell size

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Steps in Calculating (X2) value

1. Make a contingency table mentioning the frequencies in all cells.

2. Determine the expected value (E) in each cell.

3. Calculate the difference between observed and expected values in each cell (O-E).

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4. Calculate X2 value for each cell

5. Sum up X2 value of each cell to get X2 value of the table.

6. Find out p value from x2 table

7. If p > 0.05 , difference is not significant, null hypothesis

accepted;

If p < 0.05 , difference is significant , null hypothesis rejected.

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Example

• Consider a study done in a hospital where cases of breast

cancer were compared against controls from normal

population against with a family history of Ca Breast.

• 100 in each group were studied for presence of family history.

• 25 of cases and 15 among controls had a positive family

history.

• Comment on the significance of family history in breast

cancer.

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Solution

• From the numbers, it suggests that family history is 1.66

(25/15) times more common in Ca breast.

• So is it a risk factor in population?

• We need to test for the significance of this difference.

• We shall apply X2 test.

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Solution

Step – 1: Set up a null hypothesis

– H0: “There is no significant difference between incidence of

family history among cases and controls.”

Step – 2: Define alternative hypothesis

– Ha: “Family history is 1.66 times more common in Ca

breast”

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Step – 3: Calculating X2

1. Make a contingency table mentioning the frequencies in all

cells

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Step – 3: Cont..

2. Determine the expected value (E) for each cell.

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O – observed values , E – expected value

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Step – 4: Determine degree of freedom.

• DoF is determined by the formula:

DoF = (r-1) x (c-1)

where r and c are the number of rows and columns

respectively

• Here, r = c = 2.

• Hence, DoF = (2-1) x (2-1) = 1

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Step – 5: Find out the corresponding P Value

– P values can be derived by using the X2 distribution

tables

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Step – 6: Accept or reject the Null hypothesis

• In given scenario,

X2 = 3.125

• This is less than 3.84 (for P = 0.05 at dof =1)

• Hence Null hypothesis is Accepted, i.e.,

“There is no significant difference between incidence of

family history among cases and controls”

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Limitations of tests of significance

Testing of hypothesis is not decision making itself; but it helps

for decision making

Test does not explain the reasons as why the difference

exists, tests do not tell about the reason causing the difference.

Tests are based on the probabilities and as such can not be

expressed with full certainty.

Statistical inferences based on the significance tests can

not be said to be entirely correct evidences concerning

the truth of the hypothesis.

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THANK YOU

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