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Bivariate Analysis

Cross-tabulation and chi-square
So far the statistical methods we
  have used only permit us to:
• Look at the frequency in which certain
  numbers or categories occur.
• Look at measures of central tendency such
  as means, modes, and medians for one
  variable.
• Look at measures of dispersion such as
  standard deviation and z scores for one
  interval or ratio level variable.
Bivariate analysis allows us to:
• Look at associations/relationships among
  two variables.
• Look at measures of the strength of the
  relationship between two variables.
• Test hypotheses about relationships
  between two nominal or ordinal level
  variables.
For example, what does this table tell us about
        opinions on welfare by gender?
Support cutting
welfare benefits     Male           Female
for immigrants
Yes                         15                5

No                          10               20

Total                       25               25
Are frequencies sufficient to
allow us to make comparisons
        about groups?

What other information do we
           need?
Is this table more helpful?
Benefits for         Males      Female
Immigrants
Yes                  15 (60%)    5 (20%)

No                   10 (40%)   20 (80%)

Total               25 (100%) 25 (100%)
How would you write a sentence
or two to describe what is in this
             table?
Rules for cross-tabulation
• Calculate either column or row percents.
• Calculations are the number of frequencies
  in a cell of a table divided by the total
  number of frequencies in that column or
  row, for example 20/25 = 80.0%
• All percentages in a column or row should
  total 100%.
Let’s look at another example –
 social work degrees by gender
Social Work   Male                Female
Degree
BA               20 (33.3%)                20 ( %)

MSW                  30 (     )       70 (70.0%)

Ph.D.                10 (16.7%)       10 (10.0%)

                 60 (100.0%)         100 (100.0%
Questions:

What group had the largest percentage of
               Ph.Ds?

 What are the ways in which you could
     find the missing numbers?

    Is it obvious why you would use
percentages to make comparisons among
           two or more groups?
In the following table, were people with drug,
alcohol, or a combination of both most likely
     to be referred for individual treatment?
Services       Alcohol      Drugs        Both

Individual      10 (25%)    30 (60%)      5 (50%)
Treatment
Group           10 (25%)    10 (20%)      2 (20%)
Treatment
AA              20 (50%)    10 (20%)      3 (30%)

Total          40 (100%)   50 (100%)   10 (100%)
Use the same table to answer the
      following question:

   How much more likely are
  people with alcohol problems
 alone to be referred to AA than
 people with drug problems or a
combination of drug and alcohol
           problems?
We use cross-tabulation when:
• We want to look at relationships among two
  or three variables.
• We want a descriptive statistical measure to
  tell us whether differences among groups
  are large enough to indicate some sort of
  relationship among variables.
Cross-tabs are not sufficient to:
• Tell us the strength or actually size of the relationships
  among two or three variables.
• Test a hypothesis about the relationship between two or
  three variables.
• Tell us the direction of the relationship among two or more
  variables.
• Look at relationships between one nominal or ordinal
  variable and one ratio or interval variable unless the range
  of possible values for the ratio or interval variable is small.
  What do you think a table with a large number of ratio
  values would look like?
We can use cross-tabs to visually
assess whether independent and
 dependent variables might be
related. In addition, we also use
     cross-tabs to find out if
 demographic variables such as
gender and ethnicity are related
     to the second variable.
For example, gender may
   determine if someone votes
 Democratic or Republican or if
income is high, medium, or low.
  Ethnicity might be related to
where someone lives or attitudes
  about whether undocumented
 workers should receive driver’s
            licenses.
Because we use tables in these ways, we can
 set up some decision rules about how to use
                       tables.
• Independent variables should be column variables.
• If you are not looking at independent and
  dependent variable relationships, use the variable
  that can logically be said to influence the other as
  your column variable.
• Using this rule, always calculate column
  percentages rather than row percentages.
• Use the column percentages to interpret your
  results.
For example,
• If we were looking at the relationship between
  gender and income, gender would be the column
  variable and income would be the row variable.
  Logically gender can determine income. Income
  does not determine your gender.
• If we were looking at the relationship between
  ethnicity and location of a person’s home,
  ethnicity would be the column variable.
• However, if we were looking at the relationship
  between gender and ethnicity, one does not
  influence the other. Either variable could be the
  column variable.
SPSS will allow you to choose a
column variable and row variable
  and whether or not your table
   will include column or row
             percents.
You must use an additional statistic, chi-
       square, if you want to:
• Test a hypothesis about two variables.
• Look at the strength of the relationship between an
  independent and dependent variable.
• Determine whether the relationship between the
  two variables is large enough to rule out random
  chance or sampling error as reasons that there
  appears to be a relationship between the two
  variables.
Chi-square is simply an extension of a
  cross-tabulation that gives you more
   information about the relationship.
 However, it provides no information
 about the direction of the relationship
(positive or negative) between the two
                variables.
Let’s use the following table to
         test a hypothesis:
                           Education
Income         High        Low              Total
High (Above           40                             50
$40,000)
Low ($39,999                                         50
or less)
Total                 50               50           100
I have not filled in all of the information
 because we need to talk about two concepts
          before we start calculations:
• Degrees of Freedom: In any table, there are
  a limited number of choices for the values
  in each cell.
• Marginals: Total frequencies in columns
  and rows.
Let’s look at the number of choices
   we have in the previous table:
                           Education
Income         High        Low              Total
High (Above           40                             50
$40,000)
Low ($39,999                                         50
or less)
Total                 50               50           100
So the table becomes:
                           Education
Income         High        Low              Total
High (Above           40               10            50
$40,000)
Low ($39,999          10               40            50
or less)
Total                 50               50           100
The rules for determining degrees of freedom
 in cross-tabulations or contingency tables:
• In any two by two tables (two columns, two
  rows, excluding marginals) DF = 1.
• For all other tables, calculate DF as:
  (c -1 ) * (r-1) where c = columns and r =
  rows.
  ( So for a table with 3 columns and 4 rows,
  DF = ____. )
Importance of Degrees of Freedom
• You will see degrees of freedom on your SPSS
  print out.
• Most types of inferential statistics use DF in
  calculations.
• In chi-square, we need to know DF if we are
  calculating chi-square by hand. You must use the
  value of the chi-square and DF to determine if the
  chi-square value is large enough to be statistically
  significant (consult chi-square table in most
  statistics books).
Steps in testing a hypothesis:
• State the research hypothesis
• State the null hypothesis
• Choose a level of statistical significance
  (alpha level)
• Select and compute the test statistic
• Make a decision regarding whether to
  accept or reject the null hypothesis.
Calculating Chi-Square
• Formula is [0 - E]2
               E
Where 0 is the observed value in a cell
      E is the expected value in the same
      cell we would see if there was no
      association
First steps
Alternative hypothesis is: There is a relationship
  between income level and education for
  respondents in a survey of BA students.

Null hypothesis is: There is no relationship between
 income level and education for respondents in a
 survey of BA students

Confidence level set at .05
Rules for determining whether the chi-square
statistic and probability are large enough to verify a
                    relationship.
• For hand calculations, use the degree(s) of
  freedom and the confidence level you set to check
  the Chi-square table found in most statistics
  books. For the chi-square to be statistically
  significant, it must be the same size or larger than
  the number in the table.
• On an SPSS print out, the p. or significance value
  must be the same size or smaller than your
  significance level.
The formula for expected values are
             E = R*C
                           Education
Income         High        Low              Total
High (Above           25               25            50
$40,000)
Low ($39,999          25               25            50
or less)
Total                 50               50           100
Go back to our first table
                           Education
Income         High        Low              Total
High (Above           40               10            50
$40,000)
Low ($39,999          10               40            50
or less)
Total                 50               50           100
Chi-square calculation is

         Expected
         Values                     Chi-square
Cell 1   50 * 50/100            25 (40-25)2/25   9
Cell 2   50*50/100              25 (10-25)2/25   9
Cell 3   50 * 50/100            25 (10-25)2/25   9
Cell 4   50*50/100              25 (40-25)2/25   9

                                                 36

At .05, 1 = df, chi-square must be larger

than 3.84 to be statistically significant
Let’s calculate another chi-square- service
        receipt by location of residence
Service    Urban        Rural       Total

Yes        20           40          60

No         30           10          40

Total      50           50          100
For this table,
• DF = 1
• Alternative hypothesis:
  Receiving service is associated with
  location of residence.
  Null hypothesis:
  There is no association between receiving
  service and location of residence.
Calculations for chi-square are
         Expected
         Values                 Chi-square
Cell 1   50 * 60/100        30 (20-30)2/30         3.33
Cell 2   50*40/100          20 (30-20)2/20         5.00
Cell 3   50*60/100          30 (40-30)2/30         3.33
Cell 4   50*40/100          20 (10-20)2/20         5.00


                                                 16.67
At 1 DF at .01 chi-square must be greater than 6.64. Do
we accept or reject the null hypothesis?
Running chi-square in SPSS
•   Select descriptive statistics
•   Select cross-tabulation
•   Highlight your independent variable and click on the arrow.
•   Highlight your dependent variable and click on the arrow.
•   Select Cells
•   Choose column percents
•   Click continue
•   Select statistics
•   Select chi-square
•   Click continue
•   Click ok
SPSS print out


                     Chi-Square Tests

                                                  Asymp. Sig.
                         Value          df         (2-sided)
Pearson Chi-Square        2.569 a            5            .766
Likelihood Ratio          2.590              5            .763
Linear-by-Linear
                             .087            1              .768
Association
N of Valid Cases             336
  a. 2 cells (16.7%) have expected count less than 5. The
     minimum expected count is 1.57.
Recode
• To run ratio or interval level variables into SPSS
  you need to recode or change the variable into a
  categorical or nominal or ordinal variable.
You first need to decide how you will set up
  categories and assign a number to them.
For example if your ratio variables for Age are: 25,
  37, 42, 50, and 64, you might decide on two
  categories: 1 = under 50
               2 = 50 and over
Recode Instructions
•   Go to Transform menu
•   Go to Recode
•   Select different variable
•   Type in new variable name
•   Click continue
•   Enter range of ratio numbers for first category (25 to 49)
•   Enter number for first category (1) in right hand screen.
•   Click Add
•   Enter range of ratio numbers (50 to 54) for category two
•   Enter number for second category (2)
•   Click Add
•   Click Continue
•   Click Change
•   Click o.k.

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Bivariate analysis

  • 2. So far the statistical methods we have used only permit us to: • Look at the frequency in which certain numbers or categories occur. • Look at measures of central tendency such as means, modes, and medians for one variable. • Look at measures of dispersion such as standard deviation and z scores for one interval or ratio level variable.
  • 3. Bivariate analysis allows us to: • Look at associations/relationships among two variables. • Look at measures of the strength of the relationship between two variables. • Test hypotheses about relationships between two nominal or ordinal level variables.
  • 4. For example, what does this table tell us about opinions on welfare by gender? Support cutting welfare benefits Male Female for immigrants Yes 15 5 No 10 20 Total 25 25
  • 5. Are frequencies sufficient to allow us to make comparisons about groups? What other information do we need?
  • 6. Is this table more helpful? Benefits for Males Female Immigrants Yes 15 (60%) 5 (20%) No 10 (40%) 20 (80%) Total 25 (100%) 25 (100%)
  • 7. How would you write a sentence or two to describe what is in this table?
  • 8. Rules for cross-tabulation • Calculate either column or row percents. • Calculations are the number of frequencies in a cell of a table divided by the total number of frequencies in that column or row, for example 20/25 = 80.0% • All percentages in a column or row should total 100%.
  • 9. Let’s look at another example – social work degrees by gender Social Work Male Female Degree BA 20 (33.3%) 20 ( %) MSW 30 ( ) 70 (70.0%) Ph.D. 10 (16.7%) 10 (10.0%) 60 (100.0%) 100 (100.0%
  • 10. Questions: What group had the largest percentage of Ph.Ds? What are the ways in which you could find the missing numbers? Is it obvious why you would use percentages to make comparisons among two or more groups?
  • 11. In the following table, were people with drug, alcohol, or a combination of both most likely to be referred for individual treatment? Services Alcohol Drugs Both Individual 10 (25%) 30 (60%) 5 (50%) Treatment Group 10 (25%) 10 (20%) 2 (20%) Treatment AA 20 (50%) 10 (20%) 3 (30%) Total 40 (100%) 50 (100%) 10 (100%)
  • 12. Use the same table to answer the following question: How much more likely are people with alcohol problems alone to be referred to AA than people with drug problems or a combination of drug and alcohol problems?
  • 13. We use cross-tabulation when: • We want to look at relationships among two or three variables. • We want a descriptive statistical measure to tell us whether differences among groups are large enough to indicate some sort of relationship among variables.
  • 14. Cross-tabs are not sufficient to: • Tell us the strength or actually size of the relationships among two or three variables. • Test a hypothesis about the relationship between two or three variables. • Tell us the direction of the relationship among two or more variables. • Look at relationships between one nominal or ordinal variable and one ratio or interval variable unless the range of possible values for the ratio or interval variable is small. What do you think a table with a large number of ratio values would look like?
  • 15. We can use cross-tabs to visually assess whether independent and dependent variables might be related. In addition, we also use cross-tabs to find out if demographic variables such as gender and ethnicity are related to the second variable.
  • 16. For example, gender may determine if someone votes Democratic or Republican or if income is high, medium, or low. Ethnicity might be related to where someone lives or attitudes about whether undocumented workers should receive driver’s licenses.
  • 17. Because we use tables in these ways, we can set up some decision rules about how to use tables. • Independent variables should be column variables. • If you are not looking at independent and dependent variable relationships, use the variable that can logically be said to influence the other as your column variable. • Using this rule, always calculate column percentages rather than row percentages. • Use the column percentages to interpret your results.
  • 18. For example, • If we were looking at the relationship between gender and income, gender would be the column variable and income would be the row variable. Logically gender can determine income. Income does not determine your gender. • If we were looking at the relationship between ethnicity and location of a person’s home, ethnicity would be the column variable. • However, if we were looking at the relationship between gender and ethnicity, one does not influence the other. Either variable could be the column variable.
  • 19. SPSS will allow you to choose a column variable and row variable and whether or not your table will include column or row percents.
  • 20. You must use an additional statistic, chi- square, if you want to: • Test a hypothesis about two variables. • Look at the strength of the relationship between an independent and dependent variable. • Determine whether the relationship between the two variables is large enough to rule out random chance or sampling error as reasons that there appears to be a relationship between the two variables.
  • 21. Chi-square is simply an extension of a cross-tabulation that gives you more information about the relationship. However, it provides no information about the direction of the relationship (positive or negative) between the two variables.
  • 22. Let’s use the following table to test a hypothesis: Education Income High Low Total High (Above 40 50 $40,000) Low ($39,999 50 or less) Total 50 50 100
  • 23. I have not filled in all of the information because we need to talk about two concepts before we start calculations: • Degrees of Freedom: In any table, there are a limited number of choices for the values in each cell. • Marginals: Total frequencies in columns and rows.
  • 24. Let’s look at the number of choices we have in the previous table: Education Income High Low Total High (Above 40 50 $40,000) Low ($39,999 50 or less) Total 50 50 100
  • 25. So the table becomes: Education Income High Low Total High (Above 40 10 50 $40,000) Low ($39,999 10 40 50 or less) Total 50 50 100
  • 26. The rules for determining degrees of freedom in cross-tabulations or contingency tables: • In any two by two tables (two columns, two rows, excluding marginals) DF = 1. • For all other tables, calculate DF as: (c -1 ) * (r-1) where c = columns and r = rows. ( So for a table with 3 columns and 4 rows, DF = ____. )
  • 27. Importance of Degrees of Freedom • You will see degrees of freedom on your SPSS print out. • Most types of inferential statistics use DF in calculations. • In chi-square, we need to know DF if we are calculating chi-square by hand. You must use the value of the chi-square and DF to determine if the chi-square value is large enough to be statistically significant (consult chi-square table in most statistics books).
  • 28. Steps in testing a hypothesis: • State the research hypothesis • State the null hypothesis • Choose a level of statistical significance (alpha level) • Select and compute the test statistic • Make a decision regarding whether to accept or reject the null hypothesis.
  • 29. Calculating Chi-Square • Formula is [0 - E]2 E Where 0 is the observed value in a cell E is the expected value in the same cell we would see if there was no association
  • 30. First steps Alternative hypothesis is: There is a relationship between income level and education for respondents in a survey of BA students. Null hypothesis is: There is no relationship between income level and education for respondents in a survey of BA students Confidence level set at .05
  • 31. Rules for determining whether the chi-square statistic and probability are large enough to verify a relationship. • For hand calculations, use the degree(s) of freedom and the confidence level you set to check the Chi-square table found in most statistics books. For the chi-square to be statistically significant, it must be the same size or larger than the number in the table. • On an SPSS print out, the p. or significance value must be the same size or smaller than your significance level.
  • 32. The formula for expected values are E = R*C Education Income High Low Total High (Above 25 25 50 $40,000) Low ($39,999 25 25 50 or less) Total 50 50 100
  • 33. Go back to our first table Education Income High Low Total High (Above 40 10 50 $40,000) Low ($39,999 10 40 50 or less) Total 50 50 100
  • 34. Chi-square calculation is Expected Values Chi-square Cell 1 50 * 50/100 25 (40-25)2/25 9 Cell 2 50*50/100 25 (10-25)2/25 9 Cell 3 50 * 50/100 25 (10-25)2/25 9 Cell 4 50*50/100 25 (40-25)2/25 9 36 At .05, 1 = df, chi-square must be larger than 3.84 to be statistically significant
  • 35. Let’s calculate another chi-square- service receipt by location of residence Service Urban Rural Total Yes 20 40 60 No 30 10 40 Total 50 50 100
  • 36. For this table, • DF = 1 • Alternative hypothesis: Receiving service is associated with location of residence. Null hypothesis: There is no association between receiving service and location of residence.
  • 37. Calculations for chi-square are Expected Values Chi-square Cell 1 50 * 60/100 30 (20-30)2/30 3.33 Cell 2 50*40/100 20 (30-20)2/20 5.00 Cell 3 50*60/100 30 (40-30)2/30 3.33 Cell 4 50*40/100 20 (10-20)2/20 5.00 16.67 At 1 DF at .01 chi-square must be greater than 6.64. Do we accept or reject the null hypothesis?
  • 38. Running chi-square in SPSS • Select descriptive statistics • Select cross-tabulation • Highlight your independent variable and click on the arrow. • Highlight your dependent variable and click on the arrow. • Select Cells • Choose column percents • Click continue • Select statistics • Select chi-square • Click continue • Click ok
  • 39. SPSS print out Chi-Square Tests Asymp. Sig. Value df (2-sided) Pearson Chi-Square 2.569 a 5 .766 Likelihood Ratio 2.590 5 .763 Linear-by-Linear .087 1 .768 Association N of Valid Cases 336 a. 2 cells (16.7%) have expected count less than 5. The minimum expected count is 1.57.
  • 40. Recode • To run ratio or interval level variables into SPSS you need to recode or change the variable into a categorical or nominal or ordinal variable. You first need to decide how you will set up categories and assign a number to them. For example if your ratio variables for Age are: 25, 37, 42, 50, and 64, you might decide on two categories: 1 = under 50 2 = 50 and over
  • 41. Recode Instructions • Go to Transform menu • Go to Recode • Select different variable • Type in new variable name • Click continue • Enter range of ratio numbers for first category (25 to 49) • Enter number for first category (1) in right hand screen. • Click Add • Enter range of ratio numbers (50 to 54) for category two • Enter number for second category (2) • Click Add • Click Continue • Click Change • Click o.k.