Bivariate Statistics

1
Bivariate Statistics

• GTECH 201
• Lecture 17

2
Overview of Todays Topic

• Two-Sample Difference of Means Test
• Matched Pairs (Dependent Sample) Tests
• Chi-Square Goodness of Fit Test
• Kolmogorov-Smirnov Test

3
Differences Between Two Samples

• Are there significant differences between the two
  samples?
• If sample differences are significant, then
• Were can infer that the samples were drawn from
  truly different populations or vise versa
• Extending hypothesis testing
• Statistic (mean)
• Relationship between samples
• Independent
• Dependent

4
Two Sample Difference Of Means

• Numerator
• Actual difference between sample means
• Denominator
• Standard error of the difference of the means (a
  measure of expected sampling error)

5
Pooled Variance/Separate Variance

• If the population variance is equal, use PV
• If the population variances are unknown but
  assumed equal, then use modified formula
• If population variances are assumed to be
  unequal, then, use SV
• Sample variances are considered best estimators
  of population variances

6
Matched Pairs (Dependent Sample)

• One set of observations (units)
• Same location and/or same individuals
• One variable, two time periods
• Two variables, one time period
• Absence of two independent samples
• When two sets of data are collected for one group
  of observations,samples are
• Dependent
• Matched pairs difference test (is the appropriate
  inferential test)
• Each unit in the sample has two values (a matched
  pair)
• Parametric, non parametric

7
Wilcoxon Matched-pairs Signed Ranks Test

• Random sample
• Ordinal or downgraded to ordinal
• H0 ranked matched-pair differences are equal
• Test statistic with T rank sum

8
Matched Pairs t Test

• Sample are independent of each other
• In this situation, the t-test considers the
  difference between the values for each matched
  pair
• The greater the difference (d), the more
  dissimilar the results of the two values within
  the matched pair
• The mean ( ) is determined for the set of
  all matched pairs in the sample

9
Wilcoxon Rank Sum W

• Non-parametric difference of means test
• Measures magnitude of the differences in ranked
  positions

10
Goodness of Fit Tests

• Comparing an actual or observed frequency
  distribution to some expected frequency
  distribution
• Used to test the hypothesis that a a set of data
  has a particular frequency distribution
• Confirm or deny the relevance or validity of a
  particular theory
• Verify assumptions about samples

11
Chi Squared Distributions

• The total area under a chi-squared curve is equal
  to 1
• Chi-squared curve starts at 0 on the horizontal
  axis and extends indefinitely to the right,
  approaching but not touching the horizontal axis
• Chi squared curve is right skewed
• As the number of degrees of freedom become
  larger, chi-squared curves look increasingly like
  normal curves

12
c2 Function

• The above animation shows the shape of the
  Chi-square distribution as the degrees of freedom
  increase (1, 2, 5, 10, 25 and 50 )

13
Goodness of Fit Tests

• Characteristics of the expected frequency
  distribution
• Uniform or equal
• Proportional or unequal
• Normal (theoretical)
• The chi-squared statistic compares
• Observed frequency counts of a single variable
  (organized into nominal or ordinal categories)
• An expected distribution of frequency counts
  organized in the same categories

14
Rules for Using c2 Test

• Samples must be taken at random
• Variables must be organized in nominal or ordinal
  categories
• Must use absolute frequency counts
• Cannot be applied if the observations or sampling
  units are relative frequencies such as
  percentages, proportions, or rates
• If there are 2 nominal/ordinal categories, then
• both expected frequency counts must be at least
  five
• If there are 3 or more nominal/ordinal
  categories, then
• No expected frequency should be less than two
• At the most, only one-fifth of the frequency
  counts can be less than five
• This may be a reason to combine or reorganize
  categories

15
Test Statistic
Where, i 1, 2, to k (i.e., the different
categories) O is the observed frequency in a
particular category E is the expected frequency
in that same category k is the total number of
categories
16
Interpreting the value of c2

• Null and Alternative hypothesis
• Chi squared value is small, i.e., if the observed
  and expected frequencies are similar, then the
  goodness of fit is strong,
• Do not reject the null hypothesis
• Vice versa

17
Kolmogorov-Smirnov

• Goodness of fit test
• Uses data in ordinal categories, or
  interval/ratio data downgraded to ordinal
  categories
• Population is continuously distributed
• Null and Alternative hypothesis
• Cumulative relative frequencies are compared with
  cumulative frequencies expected for a normal
  distribution
• K-S test statistic (D) is the maximum absolute
  difference between two sets of cumulative values