Download Chi Square

Chi Square
2
Parametric Statistics
• Everything we have done so far assumes
that data are representative of a probability
distribution (normal curve).
• We are making inferences about the
parameters (statistics of the population) of
the distribution.
• That is why this is called parametric
statistics.
Non-Parametric Statistics
• If the data are not assumed to be part of a probability
distribution, then there is no distribution to which to
make inferences.
• The most frequent reason this happens is when data
are not interval.
• To make inferences, some characteristic of the data
must approximate a probability distribution.
• These are call non-parametric statistics.
Variables
• Nominal
– Named groups (mode)
• Ordinal
– Ordered named groups (median)
• Interval (Ratio)
– Continuous scales (mean)
A Problem
• With nominal or ordinal data we cannot compare
group means.
– average category or average of low/medium/high
• But, there ought to be some way to know if the the
distributions of responses in nominal or ordinal
categories could have occurred by chance.
• The most common way to do this is with a
 2 (chi square).
Reading Selection by Source
50
Female
Male
40
30
20
10
0
None
Book
Pre Dist
Magazine
Post Dist
Online
Contingency Tables (Cross Tabs)
Test of Independence
Book
Magazine
Online
Women
24
36
42
Men
22
19
30
• Are these numbers independent? If they are not,
then it means that one variable influences the other.
Contingency Tables (Cross Tabs)
Test of Independence
Book
Magazine
Online
Women
24
36
42
Men
22
19
30
• If gender is not influenced by reading preference
then the proportions for each category of reading
preference should be the same.
Contingency Tables (Cross Tabs)
Test of Independence
Book
Magazine
Online
Women
24
36
42
Men
22
19
30
• If reading preference is not influenced by gender
then the proportions for each category of gender
should be the same.
Contingency Tables (Cross Tabs)
Test of Independence
Book
Magazine
Online
Women
24
36
42
Men
22
19
30
• If neither is influenced by the other then the
proportions should be the same throughout the
model.
• Is the the distribution of reading preference
different by gender?
• Is the distribution of gender different by
reading preference?
• Are these two variables related?
• Null: the two variables are not related—
they are independent.
Chi Square
Tests of Independence
• Given the Observed Frequencies there
ought to be some way to imagine what the
most likely number would be in each cell if
the numbers were independent.
• This is kind of super averaging the counts
in related cells and predicting what should
be in the cell.
Contingency Tables
Test of Independence
Book
Women
24
Men
22
Magazine
?
36
?
19
?
Online
42
?
30
?
?
Chi Square
Tests of Independence
• Given the Observed Frequencies, determine
what would be in each cell if the variance in the
rows and columns were accounted for—looking
at row and column proportions simultaneously.
• This is done by:
Row total x Column total/ Total
• This new set of values is called the Expected
Frequencies
Contingency Tables
Test of Independence
Book
Women
24
Men
22
Magazine
36
27.12
42
32.43
19
18.88
Online
42.45
30
22.57
29.55
Contingency Tables
Test of Independence
Book
Women
24
Men
22
Magazine
36
27.12
42
32.43
19
18.88
Online
42.45
Row
Total
= 102
30
22.57
Column
Total = 46
29.55
Sample
Total = 173
(102 x 46)/173 = 27.12
Now What?
(Computing a Chi Square)
• The gathered data are the
Observed Values.
• Expected Values—a computation of what should
be in each cell based on the existing sample
distribution.
• First the computer builds a model that represents
the expected frequencies for each cell.
• Then the differences between the observed
frequencies the expected frequencies are
computed. (O – E)
Now What?
(Computing a Chi Square)
• Because (O – E) might be negative each
difference is squared.
• Since we want to know when the
differences are comparatively big or small
the real number difference has to be turned
into a ratio.
• Now it is time to add all of these up:
the sum of squared differences—chi square
• Last, given the df, the computed sum of
squares is compared to a distribution of
possible sum of squares.
(O – E)2
(O – E)2
E
Σ
(O – E)2
E
Chi Distribution
This curve represents the
probability of getting a given
chi square.
(The sum of each of the
squared differences divided by
the expected frequency.)
5% of the area
There is one of these for every possible degrees of freedom
df = (number of rows - 1) x (number of columns - 1)
• Sometimes the difference between the actual
values and the expected values is so small that
they it can be attributed to chance variation.
• If that is true we say that the variables are
independent.
• Sometimes the difference between the actual
values and the expected values is so large that it is
worth talking about why those differences
appeared. The difference is so large it is unlikely
to have happened by chance. (p <.05)
Contingency Tables
Test of Independence
Book
Women
24
Men
22
Magazine
36
27.12
42
32.43
19
18.88
Online
42.45
30
22.57
29.55
chi square = 1.85
Chi Distribution
This curve represents the
probability of getting a given
chi square.
(The sum of all the differences
squared divided by the total
number of data points.)
2 df
1.85
5% of the area
There is one of these for every possible degrees of freedom
(rows-1) x (columns-1)
Contingency Tables
Test of Independence
Book
Women
24
Men
22
Magazine
36
27.12
42
32.43
19
18.88
Online
42.45
30
22.57
29.55
chi square = 1.85
p = .40
 2 Cautions
• When observed values drop below 5, the
estimator has too much influence on the
statistic.
• In other words, do not do  2 with small
samples.
• Avoid over interpreting the results.
Caution: chi square only tells if the total difference was
likely to occur by chance—not individual differences.
You can only say IF the variables are related—not how.
Book
Women
24
Men
22
Magazine
36
27.12
42
32.43
19
18.88
Online
42.45
30
22.57
29.55
chi square = 1.85
p = .40
Using Excel to Compute
2
The Chi Square Calculation
• You will never see the distribution, only the
chi calculation and the p value.
• The chi table example is on the webpage
Table 1
Faculty and Student Self-Perception of Technology Competence by Gender
Reported Skill Level
Minimally Skilled
Moderately Skilled Accomplished
Male Faculty
7
56
22
Female Faculty
48
153
79
Male Students
4
15
17
Female Students
χ2 (6) = 13.22, p = .04
2
15
4
Actual counts
Degrees Chi
of Square
value
p value
Freedom
Chi Square
(Goodness of Fit)
• Special case when the expected frequencies
are predetermined.
• Are there important differences between
what we are seeing and some assumed norm
(the predetermined values)?
 2 Goodness of Fit
• Counts in categories
• Compares observed counts to norms.
• Tests to see if the differences between the
two are so large they are unlikely to have
occurred randomly.
 2 Goodness of Fit
Expected
Observed
Expected
Expected
Observed
Observed
 2 Goodness of Fit
 2 Goodness of Fit
Chi Square = 8.09
p = .018
In Excel
Ham
16
23
Cheese
34
23
Chi Square
0.01753637
=CHITEST(actual_range,expected_range)
Caprese
19
23
Examples of Goodness of Fit  2
• Technically all  2 are Goodness of fit.
• Uses might be:
– When no difference is predicted in categories.
– Comparison of two iterations of the same group
to see if change has occurred.
– Comparison to known distribution (i.e., z-scores)
Chi Square
• Non-parametric comparisons are weak.
• They serve as a motivation for parametric
analysis.
• Sometimes they are all that is possible.