Download ANOVA & Regression

ANOVA & Regression
Selecting the Correct Statistical
Test
Analysis of Variance
• Is used when you want to compare means
for three or more groups.
• You have a normal distribution (random
sample or population).
• It can be used to determine causation.
• It contains an independent variable that is
nominal and a dependent variable that is
interval/ratio.
Other properties of both t-test
and ANOVA
• Assumes equal variance (equal size or number
of observations in each group).
• Samples for both t-test and ANOVA should be
“independent” - this means that separate groups
should have different members. Memberships
should not overlap between groups.
• Calculations are based on degrees of freedom.
(You will see degrees of freedom on the SPSS
print out.
DF for t-test is n (number of observations – 1).
As with chi-square, degrees of
freedom represent:
• Ability of numbers in the data set to vary,
• DF in ANOVA is a bit more complex.
Calculations are based on the difference in
means between each group and within each
group.
• Therefore Degrees of Freedom between groups
are n (number of groups).
• Degrees of Freedom within groups are the
number of observations in each group (n) – 1,
then you add the total degrees of freedom for
each group.
For example, if we had three groups for whom we
have scores on the Depression Test
AA
30
52
24
60
19
57
45
Individual
Treatment
60
34
56
27
42
51
Group
Counseling
25
49
37
52
Degrees of Freedom
• Between Groups = (n –1) = 3 –1 = 2
• Within Groups =
Sum of (n-1) for each group
(7-1) + (6-1) + (5-1) =
6 + 5 + 4 = 15
Reading the ANOVA print-out
Report
Hi ghes t Year of School Com pleted
Race of Respondent
White
Black
Other
Total
Mean
13.06
11.89
12.47
12.88
N
1262
199
49
1510
Std.
Deviati on
2.955
2.677
4.001
2.984
ANOVA
Highes t Year of School Completed
Sum of
Squares
Between Groups
240.725
Within Groups
13195.99
Total
13436.72
df
2
1507
1509
Mean Square
120.362
8.756
F
13.746
Sig.
.000
Testing a Hypothesis with
ANOVA
• If our confidence level is .01
• Alternative Hypothesis: Ethnicity is associated
with years of education completed
• Null hypothesis: There is no association
between ethnicity and years of education
completed.
• F = 13.746 p = .000
Do we confirm or reject the null hypothesis?
Regression Analysis:
• Allows us to look at causation using two
interval/ratio variables.
• Involves predicting the value of the
dependent variable using the independent
variable. Other control variables can be
added to the regression analysis.
Calculation for Regression is
based on:
• The concept of the regression line. What points
in the association between two variables are on
or off the regression line.
• For simple or two variable regression:
y = a + bx where a = the y-intercept and b = the
slope of the line. Slope = the amount y increases
for each unit of the increase in X.
X = the x (independent variable value) used to
predict Y (dependent variable value)
Regression line when looking at association
between two variables
100000
80000
60000
40000
20000
0
0
20000
40000
Current Salary
60000
80000
100000
120000
140000
Control Variables are
Those variables that when combined with
the independent variable may affect the
value of the dependent variable.
For example when we look at the
association between beginning salary and
current salary, both age and gender may
affect salary amounts
Regression SPSS print out
Mo del Su mm ary
Model
1
R
.881a
Adjust ed
R Square
.775
R Square
.776
St d. E rror of
the Es timate
$8,096.337
a. Predic tors: (Constant), Minority Classifi cation,
Beginning Salary
ANOVA b
Model
1
Regres sion
Residual
Total
Sum of
Squares
1.1E+11
3.1E+10
1.4E+11
df
2
471
473
Mean Square
5.352E+10
65550668.16
F
816.484
Sig.
.000a
a. Predictors: (Constant), Minority Class ification, Beginni ng Salary
b. Dependent Variable: Current Sal ary
Co effi cien tsa
Model
1
(Const ant)
Beginning Salary
Mi nori ty Class ification
Unstandardized
Coeffic ient s
B
St d. E rror
2516.971
945.359
1.896
.048
-1632. 896
909.959
a. Dependent Variabl e: Current S alary
St andardiz ed
Coeffic ient s
Beta
.874
-.040
t
2.662
39.583
-1. 794
Si g.
.008
.000
.073
Let’s check on what this means about
minority classification and salary
Report
Current Sal ary
Mi nority Classifi cation
No
Yes
Total
Mean
$36023.3
$28713.9
$34419.6
N
370
104
474
Std.
Deviati on
*********
*********
*********
Hypothesis Testing:
•
•
•
Confidence level: = .05
Alternative Hypothesis is: Controlling for
minority status, beginning salary is associated
(or can predict) current salary.
Null hypothesis is. Controlling for minority
status, beginning salary is not associated (or
can predict) current salary.
Analyzing regression
Can use three values to interpret –
(1) R2 - Correlation between any independent and
control variables and the dependent variable.
(1) F – goodness of fit of the regression line.
Calculated based on the number of points off
the line.
(2) b – measure of the correlation between one
variable in the regression model and the
dependent variable. This is used when you
include multiple independent or control
variables in the model.
Hypothesis Test (continued)
Total correlation between the independent and control variables and the
dependent variables = R2 = .776 (note no p value – but the closer the
R2 is to 1.00 the better). This means that there is a high correlation
between minority classification and beginning salary combined and
current salary.
Total fit of the model to the regression line = F = 816, p. = .00 (less than
our confidence level of .05) Alternative hypothesis confirmed
Individual Beta values for beginning salary (.874 at p. = .00 and
minority status (.040 at p = .073). At p. = .05 CL only beginning
salary is statistically significant or associated with current salary.
Review of statistical tests
Statistic Test statistic
al Test

Chisquare
T-test
T
ANOVA
F
Correlat r
ion
Regres
sion
Use R2 or F for the fit of the model and b for the
correlation between each of the independent and control
variables and the dependent variable.
General rules for analyzing
results
•
The bigger the test statistic the more likely there is a relationship
between the independent and dependent variables. Values greater
than 3 are for every type of inferential statistic other than correlation
are usually statistically significant.
• Relationships can be positive or negative. You need the p value to
determine if the test statistic is actually large enough to be
statistically significant. You must always set a confidence level
before determining if the p value is large enough to be statistically
significant.
• Findings from small samples are unlikely to be significant unless
there is a very strong relationship between two variables.
How do we write up test results
• We use the test statistic and the
probability level.
• Correct procedure for professional journal
articles also requires the use of degrees of
freedom and number of observations.
• For Assignment #4 use the test statistic
and the probability level.
Proper format for this class
• The confidence level is p. = .05. Reject the null
hypothesis and accept the alternative
hypothesis. Correlation is r = .74 at p. = .04.
• The confidence level is p. = .10.Accept the null
hypothesis and reject the alternative hypothesis.
There is no association between years of
education and salary, controlling for gender; F =
.45, p. = .70.
Criteria for Using Statistical
Tests
•
•
•
•
Independent samples
Level of Measurement
Normal distribution
Sample Size (Minimum for quantitative research
should be 30)
• Robustness (can procedure be used when
basic assumptions are violated?) T-test, ANOVA,
and chi-square are considered very robust.
Research note:
• Some types of ordinal data can be used as
interval/ratio data in statistical analysis.
Montcalm and Royse state that such data should
be ranked at a least five levels, come from a
normal distribution, and result from a large
sample.
• The most common type of ordinal data used as
ratio/interval data in statistics is a likert scale.
Example of a likert scale
• 1 = Very satisfied
• 2 = Satisfied
• 3 = Neutral
• 4 = Unsatisfied
• 5 = Very unsatisfied.
Usually presented as a ranking ( 1 to 5),
implies an equal distance among the
categories.
If you do not have a random sample, it is proper to
use nonparametric statistics:
• Small sample size.
• No normal distribution or random
sampling.
• More than one mode.
• Many outliers in the data set.
• Dependent variables are ordinal or
dichotomous.
SPSS Instructions for Running
ANOVA
• Select Means
• Select One-way ANOVA
• Highlight your dependent variable (must be
ratio)
• Click on the arrow
• Highlight your factor (independent) variable
(must be nominal with at least three categories)
• Click o.k.
SPSS instructions for running
Regression
•
•
•
•
•
•
•
Select Analyze
Select Regression
Select Linear
Highlight Dependent Variable (must be ratio)
Highlight two or independent or control variables
Click on Arrow
Click o.k.
SPSS Instructions for Running
Means
•
•
•
•
•
•
Select Analyze
Select Compare Means
Select Means
Highlight Dependent (Ratio) Variable
Highlight Independent (Nominal) Variable
Click ok