Applied statistics lecture_4

Introduction to applied statistics

& applied statistical methods

Prof. Dr. Chang Zhu 1

objectives

• significance p-value

• correlation

Pearson’s r

Spearman’s rho (rs)

Kendall’s tau-b (τ)

Partial correlation

significance – p value

• Read section 2.1 in the WIKI

• Open the section Assessment

• Work with your friends to do Revision 1

3

Assessment: Revision 1


value test statistic alternative

hypothesis

null

hypothesis

p < .05

significant

accepted

rejected

p > .05

non-significant

rejected

accepted


For t-tests

•p < .05 the difference is significant.

•Look at the means of the two groups before

making decision about the direction of the

hypothesis, i.e. which group has a higher/bigger

mean?

Significance

• In general, t-values which are large in absolute magnitude are desirable;

• Values of t that are large in absolute magnitude are always associated with small p-values.

• *p<.05, **p<.01, ***p<.001 as the significance levels (the probability that the observed result arising by chance)

7

Compare

Tests of normality: p value

• Shapiro-Wilk Test of Normality

• Kolmogorov-Smirnov Test of Normality

• If p>.05, the data is normal.

• If p<.05, the data significantly deviate from

a normal distribution.

Compare

Levene’s test, p-value

Independent sample test

• Levene’s test for equality of variances

• If Levene’s p>0.05, then there is equality

of variance

• If Levene’s p<0.05, then there is no

equality of variance

– Use the corresponding row of the values for t

in the output

correlation

• A researcher is interested in the degree to

which a person spends time Facebooking

(in hours per week) and the amount of

time spent socialising with friends (number

of social encounters per month).

• He comes up with the following data set. (adapted from

http://wps.pearsoned.co.uk/ema_uk_he_dancey_statsmath

_4/84/21626/5536329.cw/index.html)

P_ID Facebook

use

Social

encounters

1 10 1 2 11 2 3 11 3 4 12 3 5 14 4 6 15 9 7 16 10

correlation

What can you predict?

correlation by scatterplot

Facebook use

(M=12.7)

deviance

from mean

squared

deviances

10 -2.7 7.29

11 -1.7 2.89

11 -1.7 2.89

12 -0.7 0.49

14 1.3 1.69

15 2.3 5.29

16 3.3 10.89

correlation

add up all the squared deviances: sum of squared errors

affected by sample size

divide by the number of participants minus 1: variance

FB_use

(M=12.7) deviance

squared

deviances

social

encounters

(M=6.14)

deviance squared

deviances

10 -2.7 7.29 1 -5.14 26.42

11 -1.7 2.89 2 -4.14 17.14

11 -1.7 2.89 3 -3.14 9.86

12 -0.7 0.49 3 -3.14 9.86

14 1.3 1.69 4 -2.14 4.58

15 2.3 5.29 9 2.86 8.18

16 3.3 10.89 10 3.86 14.90

correlation

• covariance: averaged sum

of combined deviations

correlation

• covariance: averaged sum of combined

deviations

• standardized covariance: correlation coefficient

sx: std. deviation of variable x

sy: std. deviation of variable y

correlation

SPSS output

Correlations

FB Encounters FB

Pearson Correlation 1 .900**

Sig. (2-tailed) .006

N 7 7

Encounters Pearson Correlation .900** 1

Sig. (2-tailed) .006

N 7 7

**. Correlation is significant at the 0.01 level (2-tailed).

r = .90, p < .01 (significant)

Correlation

Positive Correlation Negative Correlation

Correlation analysis

correlation

The correlation coefficient: measures the relative strength of the linear relationship between two variables

• Ranges between –1 and 1

• The closer to –1, the stronger the negative

linear relationship

• The closer to 1, the stronger the positive

linear relationship

• The closer to 0, the weaker any positive linear

relationship

A perfect positive correlation

Height

Weight

Height of A

Weight of A

Height of B

Weight of B

A linear relationship

High Degree of positive correlation

• Positive relationship

Height

Weight

r = +.80

• Moderate Positive Correlation

Weight

Shoe

Size

r = + 0.4

• Perfect Negative Correlation

Exam score

TV

watching

per

week

r = -1.0

• Negative Correlation

Exam score

TV

watching

per

week

r = -.80

• Weak negative Correlation

Weight

Shoe

Size r = - 0.2

• No Correlation (horizontal line)

Height

IQ

r = 0.0

Test of Correlations

Parametric test:

Pearson’s r is the most common correlation coefficient.

Non-parametric tests

•Spearman’s rho (rs): rank the scores, then use the

same equation as above.

•Kendall’s tau-b (τ) : taking into account tied ranks.

PRACTICE

Practice 1

Pearson’s correlation

•We collect the scores of 200 high school students on

various tests, including science, reading, and maths score,

and we want to know if there is a correlation between the

scores of each pair of the variables.

•The data file is named test_score.sav

In SPSS, choose Analyse > Correlate > Bivariate

practical guidelines page 2

SPSS output

Correlations

reading score math score science score

reading score Pearson Correlation 1 .662** .630**

Sig. (2-tailed) .000 .000

N 200 200 200

math score Pearson Correlation .662** 1 .631**

Sig. (2-tailed) .000 .000

N 200 200 200

science score Pearson Correlation .630** .631** 1

Sig. (2-tailed) .000 .000

N 200 200 200

**. Correlation is significant at the 0.01 level (2-tailed).

Practice 1

Conclusion?

Reading scores were significantly correlated with math

scores, r = .66, p < .01 (two-tailed), and science scores, r =

.63, p < .01 (one-tailed); the math scores were also correlated

with the science scores, r = .63, p < .01 (two-tailed).

(Practical guidelines page 4)

Practice 2

Partial correlation

• Use the data file Exam Anxiety.sav

• Conduct the Pearson’s correlation for the three variables:

exam, anxiety, and revise

• What is the relationship between the variable anxiety

and exam and revise


SPSS output

Correlations

Time Spent

Revising Exam

Performance (%) Exam Anxiety Time Spent Revising

Pearson Correlation 1 .397** -.709**

Sig. (2-tailed) .000 .000

N 103 103 103 Exam Performance (%)

Pearson Correlation .397** 1 -.441**

Sig. (2-tailed) .000 .000

N 103 103 103 Exam Anxiety Pearson

Correlation -.709** -.441** 1

Sig. (2-tailed) .000 .000

N 103 103 103 **. Correlation is significant at the 0.01 level (2-tailed).

Practice 2

Partial correlation

Observation:

• Exam anxiety is negatively correlated with

exam performance (r = -.441)

• Exam anxiety is also negatively correlated

with the time spent revising (revision time)

for the exam (r = -.709)

• However, exam performance is positively

related to the time spent revising (r= .397)

Practice 2

Partial correlation

• The revision time may affect the relationship between

exam anxiety and exam performance such that the more

one spends time on revision, the less anxiety one

perceives, hence better performance.

• We are capable of investigating purely the relationship

between exam anxiety and exam performance, taking

into account the effect of time spent on revising.

In SPSS, choose Analyse > Correlate > Partial

SPSS output

Correlations

Control Variables Exam Performance (%) Exam Anxiety Time Spent Revising

Exam Performance (%)

Correlation

1.000 -.247

Significance (2-tailed) . .012

df 0 100

Exam Anxiety Correlation -.247 1.000

Significance (2-tailed) .012 .

df 100 0

not controlling for time spent revising: r = -.441

Practice 2

Partial correlation

Conclusion?

Exam anxiety was significantly related to exam performance,

r = -.247, p < .05 (two-tailed), controlling for the effect of time

spent on revising.


Practice 1

•Two examiners rated the presentations of 20 students with 1

being poor and 10 meaning perfect. It is expected that the scores

would be similar.

•The data file is named presentation_rating.sav.


Practice 3

Spearman and Kendall’s tau

(nonparametric)


Practice 3

Spearman and Kendall’s tau

(nonparametric)

Conclusion?

•The rating of the two examiners was significantly correlated, rs =

.825, p < .01 (two-tailed). Or:

•The rating of the two examiners was significantly correlated, τ =

.707, p < .01 (two-tailed)


Assignment 4

• Detail:

Lecture 4_practical guidelines_assignment

(p. 7)

Deadline: November 5, 2014

Documents

Applied statistics lecture_4