Quantitatve Methods

Nonparametric Statistics
- test characteristics of populations without referring to specific parameters
- designed to test ordinal data
- when data is ordinal, the mean is not an appropriate measure of central location
- does not need data to be normally distributed
- Nonparametric is distribution free statistics
- test population locations

Wilcoxon rank sum test
- compare two populations
- samples are independent
- data are ordinal or interval
- test whether population distributions are identical or not
- in locations and shapes, spreads (variances)

Sign test
- samples are matched pairs
- compare two populations of ordinal data in a matched pair

Wilcoxon signed rank sum test
- samples are matched pairs
- compare two populations of interval data in a matched pair

Kruskal Wallis Test
- compare two or more populations
- data are ordinal or interval
- data from independent samples

Friedman test
- compare two or more populations
- data are ordinal or interval
- data from randomised block experiment

Spearman rank correlation coefficient
- test whether relationship exists between two variables
- ordinal or interval data

Analysis of Variance (ANOVA)
- compare two or more population means
- by analyzing sample variance
- of interval data
- from independent samples or blocked samples
- populations are referred to as treatments

SST - sum of square for treatment
- between treatment variation

SSE - sum of square error
- within treatment variation

MST = SST / (k-1)
MSE = SSE / (n-k)

test statistics F = MST / MSE
- if F larger than Fcritical, reject H0
- if p value less than significance, reject H0

independent samples - one way ANOVA

blocked samples - two way ANOVA

Inference
- when population variance is known or given
use Z = (x̄ - u ) /(σ/√ n)
- when population variance is NOT known
use t = (x̄ - u ) /( s/√ n)

z statistics is:

and 

when z statistics is replaced by t statistics, t statistics is:

 and

use s : sample std deviation instead of population std deviation

- t stat assume data is normal

and the confidence interval estimator of u is:

- test population proportion

and

Two Populations Inference
- assume populations are normally distributed
-test the variances are equal or not
use F-test
-if variances are equal
use t-statistics
-Else if variances are unequal
use t-statistics

Chi Squared Test
- multinomial experiment
- goodness of fit test
- nominal data
- test about population's variability

Sampling and Estimation
- central limit theorem
for a large enough sample size, the distribution of sample mean is approximately normal

the probability of Z within the significance level of α

P(-Zα/2 < Z < Zα/2) = 1 -α

Example:

The student salary distribution with mean 500, variance 10

Qn: what fraction of students earn more than 520?

Ans:

P( (x̄ - u ) /(σ) > (520 - 500)/ 10 ) = P(Z > 2) = 1 - 0.9772 = 0.0228

(for population, don't need to know n)

Qn: 93.32% of students earn less than me, how much do i earn?

Ans:     find Z from 0.9332, so Z = 1.5

    then calculate    (x̄ - 500)/ 10 = 1.5  => x̄ = 515

SPSS independent t-test and one way ANOVA test

If you want to compare means of dependent variables on a two value variable, such as gender, you use independent t-test.

In SPSS, Click Analyze -> Compare Means -> Independent Samples T Test, a screen will pop out. In the screen, select your dependent variables as test variables , select your demographic variables, such as gender as grouping variable. You need to click the define groups button and enter 1 in group1 and 2 in group2. Then click OK and the t-test will be executed, and the independent samples test table will be shown.

On the screen, look at the Levene's Test for equality of variances.
If sig value > 0.05, look at the sig (2-tailed) upper row value. If that value is  > 0.05, the data is significant.
If sig value < 0.05, look at the sig (2-tailed) lower row value. If that value is  > 0.05, the data is significant.

If you want to compare means of dependent variables on a multiple value variable, such as age group, you use one way ANOVA.

In SPSS, Click Analyze -> Compare Means -> One Way ANOVA Test, a screen will pop out. In the screen, select your dependent variables as test variables , select your demographic variables, such as age group as factor.  Then, click on Post Hoc button, and click to enable LSD.  Then, click on Options button, and click to enable Descriptive. Then clock OK and the ANOVA test will be executed, and the multiple comparisons table, and descriptive table and ANOVA table will be shown.