Docstoc

Frequency Distributions

Document Sample
Frequency Distributions Powered By Docstoc
					S
l
i
d
e           Multiple Regression
1




    Key Points about Multiple Regression

        Sample Homework Problem

       Solving the Problem with SPSS

       Logic for multiple Regression
    S
    l
    i
    d
    e   Key points about multiple regression
    2




Ø   Few, if any, phenomena in social and behavioral
    research can be explained with a single predictor.
    More realistically, social phenomena are very
    complex, requiring a number of predictors to model
    the relationship.

Ø   Multiple regression is an extension of simple linear
    regression that enables us to include multiple
    predictors in our regression equation. The
    interpretation of a multiple regression is very similar
    to the interpretation of a simple linear regression,
    but there are important differences.
    S
    l
    i
    d
    e      Similarities and differences - 1
    3




Ø   In both simple linear and multiple regression, there
    is an ANOVA test of the overall relationship.

Ø   In both simple linear and multiple regression, R2
    represents the proportion of variance explained
    (error reduced) in predicting the dependent variable
    based on the independent variable.

Ø   In both simple linear and multiple regression,
    Multiple R represents the strength of the relationship
    and the effect size. In multiple regression it is always
    positive and is not equal to any of the beta
    coefficients.
    S
    l
    i
    d
    e      Similarities and differences - 2
    4




Ø   In simple linear regression, the significance of the
    overall relationship and the relationship of each
    independent variable were the same. In multiple
    regression, there is a test of significance for the
    coefficient of each independent variable.

Ø   There is no necessary relationship between the
    significance of the overall relationship and the
    significance of the relationships for each of the
    individual predictors. When the overall relationship
    is significant, it is possible that none, some, or all of
    the individual relationships will be significant.
    S
    l
    i
    d
    e      Similarities and differences - 3
    5




Ø   Multiple regression is required to satisfy all of the
    assumptions of simple linear regression:
     Ø 1. The relationship is linear
     Ø 2. The residuals have the same variance
     Ø 3. The residuals are independent of each other
     Ø 4. The residuals are normally distributed

Ø   Plus one additional assumption:
     Ø The independent variables are independent of one
       another, i.e. they add to the variance explained
       in the dependent variable rather than explain the
       same variance explained by other independent
       variables.
    S
    l
    i
    d
    e      Similarities and differences - 4
    6



Ø   In a multiple regression equation, the coefficient for
    each individual variable represents the change in the
    dependent variable that it is uniquely responsible
    for, i.e. assuming the relationships between the
    other independent variables and the dependent
    variable.
Ø   The correlation between individual predictors results
    in contribution toward explaining the dependent
    variable made jointly by both, and not credited to
    either individual predictor.
Ø   In extreme cases, the relationship between
    independent variable is so strong that they are not
    credited with explaining the dependent variable,
    even though both might have a strong individual
    relationship to the dependent variable.
    S
    l
    i
    d
    e      Similarities and differences - 5
    7



Ø   If this happens, we may have predictors that really
    have a strong relationship having a b coefficient that
    is not statistically significant. The interpretation,
    based on the non-significant b coefficient, that the
    variable did not have a relationship would be an
    error.

Ø   To satisfy the assumption of independence of
    variable, our regression must not include variables
    that are collinear.

Ø   The diagnostic statistic for detecting
    multicollinearity is “tolerance,” which SPSS includes
    in the table of coefficients.
    S
    l
    i
    d
    e      Similarities and differences - 6
    8




Ø   In extreme cases of multicollinearity, SPSS cannot
    compute the regression equation. In this case, SPSS
    will exclude the variable which it thinks is producing
    the variable even though we have told it to include
    the variable in the analysis.
    S
    l
    i
    d
    e      Similarities and differences - 7
    9




Ø   Having more than one predictor in the regression
    equation leads to the question of which variable has
    the more important relationship to the dependent
    variable, i.e. which has the largest impact on the
    predicted scores.

Ø   Since beta coefficients are standardized, the one
    with the largest absolute value (ignoring the sign) is
    the most important, since it is the amount of
    increase in standard deviations for the dependent
    variable that is produced by a one standard deviation
    change in the independent variable.
    S
    l
    i
    d
    e

    1
        Change in response for sample size
    0



Ø   On the simple linear regression problems, the answer
    was an Incorrect application of a statistic if the
    sample size available to the analysis was less than
    the number recommended by Tabachnick and Fidell.
Ø   In reviewing problems, there were numerous
    occasions when a smaller sample yielded a
    statistically significant result, making the response
    Incorrect application of a statistic inappropriate
    itself.
Ø   For these problems, I am changing the response to
    adding a caution when the answer is true. This
    reflects the possibility that planning a sample of the
    given size risked not finding a significant result, but
    does not negate an otherwise useful result.
   S
   l
   i
   d
   e            Sample homework problem:
   1
   1
                Multiple regression – part 1
Based on information from the data set 2001WorldFactbook.sav, is the
                                 This is the general framework for the
                                              the homework
following statement true, false, problems in on multiple regression a statistic?
                                 or an incorrect application of
                                 assignment
Use .05 for alpha.               problems.



"Population growth rate" [pgrowth],"total fertility rate" [fertrate] and
"percent of the population below poverty line" [poverty] significantly
predicted "infant mortality rate" [infmort]. The relationship was strong
and reduced the error in predicting "infant mortality rate" by
approximately 75% (R² = 0.753, F(3, 91) = 92.67, p < .001).



"Population growth rate" significantly predicted "infant mortality rate", ß =
-0.393, t(91) = -4.04, p < .001. Higher values of "population growth rate"
were inversely related to lower values of "infant mortality rate".
                                      The problem includes a statement for
                                      the overall relationship, an individual
                                      statement for each of the
"Total fertility rate" significantly predicted "infant mortality rate", ß =
                                      independent variables, and a
                                      statement on
0.965, t(91) = 8.90, p < .001. Higher predictors. the relative importance were
                                      of values of "total fertility rate"
directly related to higher values of "infant mortality rate".
    S
    l
    i
    d
    e            Sample homework problem:
    1
    2
                 Multiple regression - part 2
(cont’d)

"Percent of the population below poverty line" significantly predicted
"infant mortality rate", ß = 0.280, t(91) = 4.41, p < .001. Higher values of
"percent of the population below poverty line" were directly related to
higher values of "infant mortality rate".

"Total fertility rate" [fertrate] was the most important predictor of the
value of "infant mortality rate" [infmort] compared to the other
independent variables.

o   True
o   True with caution
o   False                             The problem includes a statement for
o                                     the         relationship, an
    Incorrect application of a statistic overall for each of the individual
                                      statement
                                       independent variables, and a
                                       statement on the relative importance
                                       of predictors.
   S
   l
   i
   d
   e           Sample homework problem:
   1
   3
                   Data set and alpha
Based on information from the data set 2001WorldFactbook.sav, is the
following statement true, false, or an incorrect application of a statistic?
Use .05 for alpha.
                                            The first paragraph
                                            identifies:
"Population growth rate" [pgrowth],"total fertility rate" [fertrate] and
                                             •
                                                [poverty] to use, e.g.
"percent of the population below poverty line"The data set significantly
                                               2001WorldFactbook.sav
predicted "infant mortality rate" [infmort]. The relationship was strong
                                             • The alpha level for the
and reduced the error in predicting "infant mortality rate" by
                                               hypothesis test
approximately 75% (R² = 0.753, F(3, 91) = 92.67, p < .001).

"Population growth rate" significantly predicted "infant mortality rate", ß =
-0.393, t(91) = -4.04, p < .001. Higher values of "population growth rate"
were inversely related to lower values of "infant mortality rate".

"Total fertility rate" significantly predicted "infant mortality rate", ß =
0.965, t(91) = 8.90, p < .001. Higher values of "total fertility rate" were
directly related to higher values of "infant mortality rate".
   S
   l
   i
   d
   e           Sample homework problem:
   1
   4
                 The overall relationship
Based on information from the data set 2001WorldFactbook.sav, is the
following statement true, false, or an incorrect application of a statistic?
Use .05 for alpha.

"Population growth rate" [pgrowth],"total fertility rate" [fertrate] and
"percent of the population below poverty line" [poverty] significantly
predicted "infant mortality rate" [infmort]. The relationship was strong
and reduced the error in predicting "infant mortality rate" by
approximately 75% (R² = 0.753, F(3, 91) = 92.67, p < .001).

                            significantly predicted finding mortality rate", ß =
"Population growth rate" second paragraph states the"infant that we
                       The
                        p < to verify with multiple regression. The
-0.393, t(91) = -4.04, want .001. Higheravalues of "population growth rate"
                       finding identifies:
were inversely related to lower values of "infant mortality rate".
                         • The independent variables
                         • The dependent variable
                         • The strength of the relationship
"Total fertility rate" significantly predicted "infant mortality rate", ß =
0.965, t(91) = 8.90, p < .001. Higher values of "total fertility rate" were
directly related to higher values of "infant mortality rate".
   S
   l
   i
   d
   e           Sample homework problem:
   1
   5
                 Individual relationships
Based on information from the data set 2001WorldFactbook.sav, is the
                    true, false, or an incorrect application of a statistic?
following statementEach of the paragraphs for the individual
                   independent variables contains:
Use .05 for alpha.
                      • A statement about the significance of the
                         relationship between the individual
                         independent variable and the dependent
"Population growth        [pgrowth],"total fertility rate" [fertrate]
                    rate"variable                                  and
"percent of the population below poverty line" [poverty] significantly
                      • A statement about the direction of the
predicted "infant mortality rate" [infmort]. The relationship was strong
                        relationship between the individual
                        independent variable and the dependent
                         predicting "infant mortality rate" by
and reduced the error invariable
approximately 75% (R² = 0.753, F(3, 91) = 92.67, p < .001).

"Population growth rate" significantly predicted "infant mortality rate", ß =
-0.393, t(91) = -4.04, p < .001. Higher values of "population growth rate"
were inversely related to lower values of "infant mortality rate".

"Total fertility rate" significantly predicted "infant mortality rate", ß =
0.965, t(91) = 8.90, p < .001. Higher values of "total fertility rate" were
directly related to higher values of "infant mortality rate".
    S
    l
    i
    d
    e             Sample homework problem:
    1
    6
                    Importance of variables
"Percent of the population below poverty line" significantly predicted
"infant mortality rate", ß = 0.280, t(91) = 4.41, p < .001. Higher values of
                             The last paragraph is a statement of the
                             relative importance of were directly
"percent of the population below poverty line" the predictors, related to
                             e.g. which variable makes the largest
higher values of "infant mortality rate".dependent variable.
                             change in the


"Total fertility rate" [fertrate] was the most important predictor of the
value of "infant mortality rate" [infmort] compared to the other
independent variables.
                         The answer will be
                         True if all parts of
                         the problem are
o   True                 correct.
o   True with caution                                The answer will be
                                                     False if any part of the
o   False                                            problem is not correct.
o   Incorrect application of a statistic

The answer to a problem                         The answer to a problem will
will be True with caution if                    Incorrect application of a
the analysis includes an                        statistic if the level of
ordinal or we do not meet                       measurement or multicollinearity
the sample size                                 requirement is violated.
requirement.
S
l
i
d
e   Solving the problem with SPSS:
1
7
         Level of measurement




           Multiple regression requires that the dependent
           variable be interval and the independent variables be
           interval or dichotomous. "Infant mortality rate"
           [infmort] is interval level, satisfying the requirement
           for the dependent variable. "Population growth rate"
           [pgrowth] is interval level, satisfying the requirement
           for the independent variable. "Total fertility rate"
           [fertrate] is interval level, satisfying the requirement
           for the independent variable. "Percent of the
           population below poverty line" [poverty] is interval
           level, satisfying the requirement for the independent
           variable.
S
l
i
d
e   Solving the problem with SPSS:
1
8
         Multiple regression -1
                        Before we can address
                        the other issues involved
                        in solving the problem,
                        we need to generate the
                        SPSS output.




                     Select Regression
                     > Linear… from the
                     Analyze menu.
S
l
i
d
e   Solving the problem with SPSS:
1
9
         Multiple regression -2
                            First, move the
                            dependent variable
                            infmort to the
                            Dependent list
                            box.




                        Second, move the independent
                        variables pgrowth, fertrate, and
                        poverty to the Independents
                        list box.




                     Third, click on the
                     Statistics button to add
                     the additional
                     statistics.
S
l
i
d
e   Solving the problem with SPSS:
2
0
         Multiple regression -3




                                  Second, click on the
                                  Continue button to
                                  close the dialog box.
     First, in addition to the
     SPSS defaults, we add
     the check box for
     Descriptives and
     Collinearity diagnositics.
S
l
i
d
e   Solving the problem with SPSS:
2
1
         Multiple regression -4




                             When we return to the
                             Linear Regression
                             dialog box, we click on
                             OK to obtain the
                             output.
S
l
i
d
e   Solving the problem with SPSS:
2
2
           Multicollinearity




          The tolerance values for all of the independent
          variables are larger than 0.10: "population growth
          rate" [pgrowth] (0.287), "total fertility rate" [fertrate]
          (0.230) and "percent of the population below poverty
          line" [poverty] (0.673).

          Multicollinearity is not a problem in this regression
          analysis.
S
l
i
d
e   Solving the problem with SPSS:
2
3
              Sample size

             Using the rule of thumb from Tabachnick and Fidell
             that the required number of cases should be the
             larger of the number of independent variables x 8
             + 50 or the number of independent variables +
             105, multiple regression requires 108 cases. With
             95 valid cases, the sample size requirement is not
             satisfied. A caution should be added to our findings.




                          NOTE: adding a caution to our
                          findings rather than concluding that it
                          is not an appropriate use of statistics
                          is a more reasonable response than
                          what we did for multiple regression.
          S
          l
          i
          d
          e       Solving the problem with SPSS:
          2
          4
              Interpreting the overall relationship - 1

The first sentence in the
finding states that:                                                     The R² of .753 is the
"Population growth rate"                                                 reduction in error
[pgrowth],"total fertility                                               achieved by using scores
rate" [fertrate] and "percent                                            for Population growth
of the population below                                                  rate" [pgrowth],"total
poverty line" [poverty]                                                  fertility rate" [fertrate]
significantly predicted                                                  and "percent of the
"infant mortality rate"                                                  population below poverty
[infmort]. The relationship                                              line" [poverty] to predict
was strong and reduced                                                   scores for "infant
the error in predicting                                                  mortality rate" [infmort].
"infant mortality rate" by
approximately 75% (R²
= 0.753, F(3, 91) =
92.67, p < .001).




                                The overall relationship between the independent
                                variables "population growth rate"
                                [pgrowth],"total fertility rate" [fertrate] and
                                "percent of the population below poverty line"
                                [poverty] and the dependent variable "infant
                                mortality rate" [infmort] was statistically
                                significant, R² = 0.753, F(3, 91) = 92.67, p <
                                .001.
         S
         l
         i
         d
         e       Solving the problem with SPSS:
         2
         5
             Interpreting the overall relationship - 2

The first sentence in the
finding states that:
"Population growth rate"
[pgrowth],"total fertility
rate" [fertrate] and
"percent of the population
below poverty line"
[poverty] significantly
predicted "infant mortality
rate" [infmort]. The
relationship was strong and
reduced the error in
predicting "infant mortality
rate" by approximately 75%
(R² = 0.753, F(3, 91) =
92.67, p < .001).




                                 We reject the null hypothesis that all of
                                 the partial slopes (b coefficients) = 0 and
                                 conclude that at least one of the partial
                                 slopes (b coefficients) ≠ 0.
          S
          l
          i
          d
          e       Solving the problem with SPSS:
          2
          6
              Interpreting the overall relationship - 3

The first sentence in the
finding states that:
"Population growth rate"
[pgrowth],"total fertility
rate" [fertrate] and "percent
of the population below
poverty line" [poverty]
significantly predicted
"infant mortality rate"
[infmort]. The relationship
was strong and reduced
the error in predicting
"infant mortality rate" by
approximately 75% (R² =
0.753, F(3, 91) = 92.67, p
< .001).



                                  The Multiple R of 0.868 was correctly
                                  characterized as a strong relationship,
                                  using Cohen’s criteria:

                                        • r < .1 =   Trivial
                                        • .1 ≤ r <   .3 = Small or weak
                                        • .3 ≤ r <   .5 = Medium or
                                        moderate
                                        • r ≥ .5 =   Large or strong
    S
    l
    i
    d
    e       Solving the problem with SPSS:
    2
    7
        Interpreting individual relationships - 1
   The second sentence in the finding states that:
   "Population growth rate" significantly
   predicted "infant mortality rate", β = -0.393,
   t(91) = -4.04, p < .001. Higher values of
   "population growth rate" were inversely related to
   lower values of "infant mortality rate".




The individual relationship
between the independent
variable "population growth
rate" [pgrowth] and the
dependent variable "infant
mortality rate" [infmort] was
statistically significant, β = -
0.393, t(91) = -4.04, p <
.001.




                                   We reject the null hypothesis that the partial slope
                                   (b coefficient) for the variable "population growth
                                   rate" = 0 and conclude that the partial slope (b
                                   coefficient) for the variable "population growth rate"
                                   ≠ 0.
S
l
i
d
e        Solving the problem with SPSS:
2
8
     Interpreting individual relationships - 2
The second sentence in the finding states that:
"Population growth rate" significantly predicted
"infant mortality rate", β = -0.393, t(91) = -4.04,
p < .001. Higher values of "population
growth rate" were inversely related to lower
values of "infant mortality rate".




                            The negative sign of the B coefficient and the
                            Beta coefficient implies that higher values of
                            "population growth rate" were inversely related
                            to lower values of "infant mortality rate".
   S
   l
   i
   d
   e        Solving the problem with SPSS:
   2
   9
        Interpreting individual relationships - 3
  The third sentence in the finding states that:
  "Total fertility rate" significantly predicted
  "infant mortality rate", β = 0.965, t(91) =
  8.90, p < .001. Higher values of "total fertility
  rate" were directly related to higher values of
  "infant mortality rate".




The individual relationship
between the independent variable
"total fertility rate" [fertrate] and
the dependent variable "infant
mortality rate" [infmort] was
statistically significant, β = 0.965,
t(91) = 8.90, p < .001.




                                        We reject the null hypothesis that the partial slope
                                        (b coefficient) for the variable "total fertility rate"
                                        = 0 and conclude that the partial slope (b
                                        coefficient) for the variable "total fertility rate" ≠
                                        0.
S
l
i
d
e        Solving the problem with SPSS:
3
0
     Interpreting individual relationships - 4
The third sentence in the finding states that:
"Total fertility rate" significantly predicted "infant
mortality rate", β = 0.965, t(91) = 8.90, p <
.001. Higher values of "total fertility rate"
were directly related to higher values of
"infant mortality rate".




              The positive sign of the B coefficient and the
              Beta coefficient implies that higher values of
              "total fertility rate" were directly related to
              higher values of "infant mortality rate".
      S
      l
      i
      d
      e       Solving the problem with SPSS:
      3
      1
          Interpreting individual relationships - 5
     The fourth sentence in the finding states that:
     "Percent of the population below poverty
     line" significantly predicted "infant mortality
     rate", β = 0.280, t(91) = 4.41, p < .001.
     Higher values of "percent of the population below
     poverty line" were directly related to higher
     values of "infant mortality rate".




The individual relationship
between the independent variable
"percent of the population below
poverty line" [poverty] and the
dependent variable "infant
mortality rate" [infmort] was
statistically significant, β =
0.280, t(91) = 4.41, p < .001.




                                     We reject the null hypothesis that the partial slope
                                     (b coefficient) for the variable "population growth
                                     rate" = 0 and conclude that the partial slope (b
                                     coefficient) for the variable "population growth rate"
                                     ≠ 0.
 S
 l
 i
 d
 e        Solving the problem with SPSS:
 3
 2
      Interpreting individual relationships - 6
The fourth sentence in the finding states that:
"Percent of the population below poverty line"
significantly predicted "infant mortality rate",
β = 0.280, t(91) = 4.41, p < .001. Higher
values of "percent of the population
below poverty line" were directly related
to higher values of "infant mortality
rate".




              The positive sign of the B coefficient and the
              Beta coefficient implies that higher values of
              "percent of the population below poverty line"
              were directly related to higher values of "infant
              mortality rate".
  S
  l
  i
  d
  e       Solving the problem with SPSS:
  3
  3
      Interpreting individual relationships - 7

The fifth sentence in the finding states
that:
"Total fertility rate" [fertrate] was the most
important predictor of the value of "infant
mortality rate" [infmort] compared to the
other independent variables.




                            "Total fertility rate" [fertrate] was the most
                            important predictor because the absolute value
                            of it's beta coefficient (0.965) was larger than
                            the absolute value of the beta coefficients for
                            the other independent variables.
S
l
i
d
e            Solving the problem with SPSS:
3
4
                 Answering the question


    The findings for this problem state that:
      • "Population growth rate" [pgrowth],"total fertility rate" [fertrate] and
          "percent of the population below poverty line" [poverty] significantly
          predicted "infant mortality rate" [infmort]. The relationship was
          strong and reduced the error in predicting "infant mortality rate" by
          approximately 75% (R² = 0.753, F(3, 91) = 92.67, p < .001).
      • "Population growth rate" significantly predicted "infant mortality
          rate", ß = -0.393, t(91) = -4.04, p < .001. Higher values of
          "population growth rate" were inversely related to lower values of
          "infant mortality rate".
      • "Total fertility rate" significantly predicted "infant mortality rate", ß =
          0.965, t(91) = 8.90, p < .001. Higher values of "total fertility rate"
          were directly related to higher values of "infant mortality rate".
      • "Percent of the population below poverty line" significantly predicted
          "infant mortality rate", ß = 0.280, t(91) = 4.41, p < .001. Higher
          values of "percent of the population below poverty line" were directly
          related to higher values of "infant mortality rate".
      • "Total fertility rate" [fertrate] was the most important predictor of
          the value of "infant mortality rate" [infmort] compared to the other
          independent variables.

                                               All of the statements of findings
                                               are true, so the answer to the
                                               question is True with caution.
                                               The caution is added because we
                                               did not satisfy the required sample
                                               size.
S
l
i
d
e              Logic for multiple regression:
3
5
                  Level of measurement



                             Measurement
                                level of
                             independent
                               variable?
             Nominal                           Interval/Ordinal
                                                /Dichotomous


          Inappropriate
          application of                       Measurement
            a statistic                           level of
                                                dependent
                                                 variable?
                            Interval/ordinal                         Nominal/
                                                                   Dichotomous


Strictly speaking, the                                            Inappropriate
test requires an interval                                         application of
level variable. We will                                             a statistic
allow ordinal level
variables with a
caution.
S
l
i
d
e   Logic for multiple regression:
3
6
           multicollinearity


               Compute linear
             regression including
             descriptive statistics




              Tolerance for all
                independent
              variables ≥ 0.10?
                                             No



                         Yes          Inappropriate
                                      application of
                                        a statistic
   S
   l
   i
   d
   e              Logic for multiple regression:
   3
   7
                    Sample size requirement


                                   Compute linear
                                 regression including
                                 descriptive statistics




                                     Valid cases
                                 satisfies computed
                                    requirement?
                                                                  No


The sample size requirement is               Yes          Caution added
the larger of :                                            to any true
                                                             findings
 • the number of independent
   variables x 8 + 50

 • the number of independent                              NOTE: violation of
   variables + 105                                        sample size
                                                          requirements is a
                                                          caution rather than an
                                                          inappropriate application
                                                          of a statistic.
S
l
i
d
e          Logic for multiple regression:
3
8
    Significant, non-trivial overall relationship


                     Probability for F-test
                     for all coefficients
                     less than or equal to
                     alpha?
                                                    No


                               Yes                False




                  Effect size (Multiple R) is
                  not trivial by Cohen’s scale,
                  i.e. equal to or larger than
                  0.10?
                                                           No


                               Yes                        False
S
l
i
d
e    Logic for multiple regression:
3
9
    Strength of overall relationship


           Strength of relationship
           correctly interpreted
           (Multiple R)?
                                               No


                        Yes                   False




              Reduction in error
              correctly interpreted
              based Multiple R²?
                                       No


                        Yes           False
S
l
i
d
e   Logic for multiple regression: Significance
4
0
      and direction individual relationships


                     Probability for t-test
                     for B coefficient less
                     than or equal to
                     alpha?
                                                 No
These steps must
be repeated for
each independent               Yes             False
variable.



                   Direction of relationship
                   correctly interpreted
                   based on B or Beta
                   coefficient?
                                                       No


                               Yes                 False
S
l
i
d
e         Logic for multiple regression:
4
1
        Importance of individual predictors


                              Predictor with largest
                              absolute Beta
                              identified as most
                              important?
                                                         No


                                        Yes            False




                          The statistics in the SPSS
                          output match all of the
                          statistics cited in the
                          problem?
                                                                No


    Add caution if
                                        Yes                    False
    dependent or
    independent variable is
    ordinal or we do not
    meet sample size                   True
    requirement.

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:0
posted:7/23/2013
language:English
pages:41