Is the Association Statistically Significant Session 16

1
Is the Association Statistically Significant
Session 16
2
Tests of Statistical Significance

• Nominal
• Lambda t test
• Phi ?2
• Contingency Coefficient ?2
• Cramers V ?2
• Ordinal
• Gamma t test
• Somers d t test
• Tau-b, Tau-c t test
• Interval
• Pearsons r t test

3
Chi-Square TestStatistical Significance for
Nominal and Ordinal Level Variables

• To be used when
• Variables are not interval level
• Not normally distributed

4
One-Way Chi-Square Distribution of Values Across
a Single Variable

• Are variations across cell frequencies chance
  variations or are they a pattern?
• Null hypothesis is that there is no pattern of
  distribution. In other words, cases are
  distributed evenly across cells.
• Alternative hypothesis is that categories vary.
  Distribution is not the same across all categories

5

• We have two sets of cell frequencies
• The cell frequencies that correspond with the
  null hypothesis
• The observed cell frequencies
• How large is the discrepancy between these two
  sets of values?

6
Formula

• ?2 S(fo fe)2
• fe
• The closer fo is to fe, the smaller the value of
  the chi-square test
• The larger the discrepancy, the larger the value
  of the chi-square test
• A larger value means we are more likely to reject
  the null and say that there is a pattern

7

• degrees of freedom are
• k 1
• where k the number of categories

8
Two-Way Chi-Square Distribution of Values Across
a Two Variables

• Used to compare two frequency distributions in
  other words, a crosstab
• Null hypothesis is that cases are distributed
  evenly across cells.
• Alternative hypothesis is that there is variation
  in the distribution of values of one variable
  across categories of the other

9
Formulas and Calculations

• Expected frequency for null hypothesis is based
  on marginal values
• Formula for the chi-square test is the same
• Degrees of freedom
• df (r 1)(c 1)

10

• Median test (p. 302-305) skip this

11
Linear Regression

• Provides a way to evaluate the influence of one
  independent variable on the dependent variable,
  controlling for the influence of other variables

12
Statistics Produced by Regression

• a the constant
• y-hat the predicted values of y given certain
  values of the independent variables
• e the error, the discrepancy between the actual
  observed value of y and the predicted value of y,
  the slush factor
• beta coefficients the influence on y of a one
  unit change in x
• standardized beta coefficients puts the
  independent variables in the same metric

13
Statistics Produced by Regression

• t tests the statistical significance of x on y
• p values the probability of the observed value
  of the beta coefficient if the true influence
  were 0
• R-squared how well the model fits the data, or,
  the percentage of variation in y explained by the
  variation in the independent variables

14

• The Adjusted R-squared or coefficient of multiple
  determination Collectively, urbanization,
  population growth, and GDP explain 79 of
  variation in female literacy rate.

15

• For each one percent increase in the percentage
  of people living in cities, female literacy
  increases by .61 or six tenths of one percent
• For each one percent increase in the annual
  population the female literacy rate decreases by
  13.7 percent.
• Gross domestic product per capita does not have a
  statistically significant influence on female
  literacy

16
Why Use Multivariate Analysis - Regression?

• Descriptive Statistics one variable
• Measures of Association two variables
• Multivariate Analysis three variables or more

17
Why Multivariate Analysis Regression

• To identify spurious relationships
• To correctly specify relationships
• To thoroughly describe a process
• the goal of research is to explain why variables
  vary

18
The Basic Regression Model

• Y a bX e
• Y is the observed value of the dependent variable
• a is the expected value of Y when X 0 (a
  baseline value)
• b is the slope steep when X has a strong
  influence on Y (in other words, b is larger)
• e is the amount of variation in Y that cant be
  explained by X

19
The Regression Line?

• The regression line (the slope) is drawn to
  minimize the distance between the slope and the
  plotted points which are the observed values of
  the dependent variable
• The regression line represents predicted values
  (predicted by the equation) and the points
  represent actual observed values

20

• Using the slope coefficient (b), the actual
  values of X, and the value of a, we can plot the
  regression line.
• The error term, e, is the distance between the
  regression line (the slope) and the location of
  the actual observed points, the values of the
  dependent variable.

21
Assumptions for Regression

• Both the independent and the dependent variables
  are measured at the interval level
• The relationship is linear
• Variables must be normally distributed or sample
  must be large
• Sample must be random for tests of statistical
  significance

22
The Significance of the Errors

• Back to the proportionate reduction in error
• The errors should be as small as possible
• We use the average value of Y to guess
• We compare this to our guess using the value of X
  for that observation

23
Pearsons, Regression, and the Coefficient of
Determination

• Coefficient of Determination is also know as the
  R-squared. And if there is more than one
  independent variable its the adjusted R-squared.
  Enough names for ya? The adjusted R-squared
  takes the number of independent variables into
  consideration. Kind of like degree of difficulty
  in diving and gymnastics.

24

• The R-squared value is the percent of variation
  in the dependent variable that is explained by
  the independent variables, collectively.

25
The Other Statistics

• T-score and p-value the statistical
  significance of the individual coefficients. In
  other words, is the influence of this independent
  variable on the dependent variable statistically
  different from 0?
• The beta coefficient the magnitude of the
  influence of X on Y. The amount of change in Y
  for a one unit change in X.