Chapter 9'1: Using Chisquare analysis to analyze the fit of a proposed model

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Chapter 9.1 Using Chi-square analysis to analyze
the fit of a proposed model

• Goals
• 1. For a Chi-square Goodness of Fit test
• a. Know when to use it.
• b. Know the assumptions.
• c. Be able to perform all 9 steps.
• d. Know how to make a statistical conclusion and
  an interpretation of the results.
• 2. Be able to calculate the expected counts
  for a cell.
• 3. Be able to calculate the chi-square value for
  a cell as well as the chi-square value for the
  model.

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how to test one sample proportion vs. some
expected proportion

• Research question
• We are interested in simultaneously estimating
  multiple unknown proportions for a population
  based on a sample.
• Example
• From newspaper articles, we have read that the US
  is approximately 55 Caucasian, 20 Hispanic, 15
  African-American, and 10 Other races. We want to
  test if the University of Wyoming follows these
  percentages.

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Multiple comparisons?

• We could perform 4 studies, each testing one of
  the proportions
• Test1 Is the proportion Caucasian .55?
• Test2 Is the proportion Hispanic .20? Etc.
• But this introduces the problem of multiple
  comparisons, and it doesnt really test what we
  want.
• We want one OVERALL test to answer Does this
  model (in its entirety) fit the data?
• We want an overall test to look for any
  differences among all the parameters in which we
  are interested. We want something to analyze a
  Goodness of fit for the model.

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To answer this question, we will use the
following ideas

• We will set up our actual data in a table with
  r rows and c columns.
• Using the null hypothesis, we will compute
  another table for the expected counts for each
  cell.
• We will compare each actual value vs. its
  corresponding expected value.
• If these differences (in total) are large, then
  we will reject the model.
• If these differences (in total) are small, then
  we will not reject the model.

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Calculating expected counts

• Using our problem above, we have heard that the
  population proportions should be
• Caucasian Hispanic African-American Other
• 55 20 15 10
• Let our sample size be equal to 500. Our sample
  results are 440, 15, 20, and 25. For each race,
  how many people do we expect if our null
  hypothesis is true?

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Observed and expected Frequencies
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Chi Square
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Chi Square cont.
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Example Conduct a Hypothesis test to see if the
students at UW follow the percentages 55/ 20/ 15/
10 using alpha .05.

• Step 1. Hypotheses
• The population proportions for Caucasian/
  Hispanic/ African-American/Other are .55, .20,
  .15, and .10
• There is at least one difference in the
  population proportions
• Step 2. Find critical Chi Square for alpha .05
• Step 3. Collect data

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Checking assumptions

• Step 4. Assumptions
• a. Data is collected with a SRS
• Reason want our data to be representative
• b. Population (N) is at least 10 times sample
  size (n)
• Reason we want the probability of selecting a
  yes to be independent from person to person,
  thus the probability of a yes is constant.
• c. No more than 20 of the Expected cell counts
  are less than 5. All expected cell counts are at
  least 1.
• -We want to get an accurate estimate of the true
  proportion.
• Step 5. Calculate the test statistic.

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Chi Square Calculations
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Chi Square Calculations, cont.
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Further Steps

• Step 6. p-Value from the table,
• the p-value is lt .0005
• (since it is off the chart)
• 7. Comparison
• pltalpha
• 8. Statistical Conclusion
• reject the null hypothesis
• 9. Interpretation
• There is significant evidence that the true model
  for race is different than 55 Caucasian/ 20
  Hispanic/15 African American /10 Other, in the
  population, alpha .05.

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Final notes

• Note If our data supported a Fail to reject
  the null hypothesis decision, then our
  interpretation would be the same but just add
  NOT.
• Specifically, we would sayThere is not
  significant evidence that the truemodel for race
  is different than 55 Caucasian/ 20 Hispanic/15
  African American /10 Other, in the population,
  alpha .05.
• Note The actual counts for a table must be an
  integer.
• The expected counts for a table DO NOT have to be
  an integer
• If we expected 55 of the sample to be Caucasian,
  and
• our sample size 50 people, then our expected
  counts
• for that cell would be (.55)(50) 27.5 people

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Chi square test of independence

• New Situation
• We are interested in simultaneously estimating
  multiple unknown proportions and comparing these
  multiple proportions to see if they are all the
  same or if at least one of them is different.
• Example Cancer rates for different cities in the
  US (testing if multiple proportions are the
  same)
• We want to investigate the claim that different
  cities have different rates of cancer. Below is a
  breakdown of the number of households affected by
  cancer for six cities. A household is either
  affected or not affected. We would like to know
  if the proportions of affected households in the
  different cities are the same or if at least one
  of them is different than the rest.

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Data
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What does this mean?

• We want one OVERALL test to answer Are all of
  these proportions the same?
• We want an overall test to look for any
  differences among all the parameters in which we
  are interested. We want something to analyze a
  Goodness of fit for the model of all
  proportions being equal.

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To answer this question, we will use the
following ideas

• We will set up our actual data in a table with
  r rows and c columns.
• Using the null hypothesis, we will compute
  another table for the expected counts for each
  cell.
• We will compare each actual value vs. its
  corresponding expected value.
• If these differences (in total) are large, then
  we will reject the model.
• If these differences (in total) are small, then
  we will not reject the model.

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Differences from goodness of fit test

• Using our problem above, we know that if cancer
  rates are the same, then the population
  proportions should be the same for each of the
  six cities.
• Note We are not saying that we know that the
  population proportion is equal to some value like
  
• p .08. We are only saying that whatever that
  value is, that it is the same for all cities.
• Also note the difference between this statement
  and the one from last lecture
• (The proportions are equal to .55, .20, .15, .10
  for Caucasian, Hispanic, African-American and
  Other)

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Row and column totals
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Calculating expected values
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Calculating Chi Square
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Example Conduct a Hypothesis test to see if the
cancer rates for the six cities are the same,

• Step 1. Hypotheses
• H0 The population proportions for being
  affected by cancer are the same for the six
  cities
• H1 There is at least one difference in the
  population proportions
• These hypotheses can also be written with
  symbols
• H1 at least one ? is different from the rest

H0
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How do we calculate Degrees of freedom?

• df number of cells that are free and not
  pre-determined
• So for us, df (r-1)(c-1) (6-1)(2-1)
• df 5

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Data for cancer study
Observed
Expected
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Calculations
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Calculations, cont.
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Deciding if result is significant

• 6. p-Value From the table, the p-value is
  between .01 and .02
• 7. Comparison
• p lt alpha
• 8. Statistical Conclusion
• Reject the null hypothesis
• 9. Interpretation
• There is significant evidence that the
  proportions for getting cancer in the 6 cities
  are different in thepopulation, alpha .05.
• Note If our data supported a Fail to reject
  the null hypothesis decision, then our
  interpretation would be the same but just add
  NOT.
• Specifically, we would say There is not
  significant evidence that the proportions for
  getting cancer in the 6 cities are different in
  the population, alpha .05.

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Analyzing which cell contributes the most to a
significant overall chi-square value

• Look at the individual chi-square values for each
  cell and find the largest.
• This is the cell in which observed and expected
  are the most different.
• Interpretation for our data above
• The chi-square value for the cell corresponding
  to city4 and affected with cancer is 11.338.
  
• For this cell, the observed value of 12 was much
  greater than the expected value of 4.7 people.

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Testing if 2 variables are independent

• We want to test if the variables city and
  cancer status are related.
• That is, are the variables city and cancer
  status independent or dependent?
• Does knowing the value for city help us in
  determining the proportion of individuals
  affected with cancer?
• If the variables are independent, then the
  proportion of people with cancer should be the
  same for all cities. Thus, by knowing the city,
  this does NOT help us out.
• If the variables are dependent, then the
  proportion of people with cancer should NOT be
  the same for all cities. Thus, by knowing the
  city, this DOES help us out.