Chi-squared Tests
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Chi-squared Tests. For testing significance of patterns in qualitative data Test statistic is based on counts that represent the number of items that fall in each category Test statistics measures the agreement between actual counts and expected counts assuming the null hypothesis.

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For testing significance of patterns in qualitative data

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For testing significance of patterns in qualitative data

Chi-squared Tests

  • For testing significance of patterns in qualitative data

  • Test statistic is based on counts that represent the number of items that fall in each category

  • Test statistics measures the agreement between actual counts and expected counts assuming the null hypothesis


For testing significance of patterns in qualitative data

Chi-squared Distribution

The chi-square distribution can be used to see whether or not an observed counts agree with an expected counts.Let

O = observed count and

E = Expected count


For testing significance of patterns in qualitative data

Testing if Observed Counts

are in Agreement with Known Percentages

Consider items of a population distributed over k categories in in proportions

If H0 is true then we expect

Ei = n , expected frequency

for the ith category as opposed to Oi, observed frequency.


For testing significance of patterns in qualitative data

An Example

Biased Coin?

ObservedExpected

FrequencyFrequency

H4050

T6050

sum100100


For testing significance of patterns in qualitative data

degrees of freedom = (R –1)(C – 1)

R = number of rows

C = number of columns


For testing significance of patterns in qualitative data

Is our chi square value an extreme outcome just by chance while in fact the null hypothesis is true and sample frequencies are not significantly apart from the ideal frequencies?

Note that chi-squared statistic is a positive number


For testing significance of patterns in qualitative data

  • only the right-hand sideof the table is used

  • nondirectional test

  • the statistic has no sign


For testing significance of patterns in qualitative data

ObservedExpected

DieFrequencyFrequency

1410

2610

31710

41610

5810

6910

sum6060


For testing significance of patterns in qualitative data

degrees of freedom =

number of terms -1


For testing significance of patterns in qualitative data

2 x 2 contingency tables

Chi-squared test for independence

Var B

total

b1

b2

Var A

a1

a2

total

Ho : The two variable are independent

Ha : The two variables are associated


For testing significance of patterns in qualitative data

Result

notdef.

total

def

Operator

A

100

900

1000

B

60

440

500

total

160

1340

1500


For testing significance of patterns in qualitative data

Result

notdef.

total

def

Operator

A

100

900

1000

B

60

440

500

total

160

1340

1500

Total number of items=1500

Total number of defective items=160

Overall defective rate =160/1500=0.1067

Now, apply this rate to the number of items produced by each operator.


For testing significance of patterns in qualitative data

Result

notdef.

total

def

Operator

A

100

900

1000

B

60

440

500

total

160

1340

1500

Expected defective from Operator A

= 1000 * 0.1067 = 106.7

(expected not defective=1000-106.7=893.3)

Expected defective from Operator B

= 500 * 0.1067 = 53.3

(expected not defective=500-53.3=446.7)


For testing significance of patterns in qualitative data

notdef.

total

def

Operator

1000

A

100

900

B

60

440

500

total

160

1340

1500

Expected

notdef.

total

def

Operator

A

106.7

893.3

B

53.3

446.7

total

Result


For testing significance of patterns in qualitative data

r x c contingency tables

SAANODSD

Gr 112184812

Gr2482210810

Gr3104121012


For testing significance of patterns in qualitative data

  • use when you have categorical data

  • measure the difference between actual counts and expected counts

  • test the independence of two variables

  • Assumptions:data set is a random sampleyou have at least 5 counts in each category

  • degrees of freedom =(categories var1 -1)(categories var2 -1)


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