ANOVA Test compares three or more groups or Analysis of ...
Transcript of ANOVA Test compares three or more groups or Analysis of ...
1
Analysis of Variance (ANOVA)
Two types of ANOVA tests: Independent measures and Repeated measures
ANOVA Test compares three or more groups or conditions. Focuses on variances instead of means.
Comparing 2 means:
Comparing 3 means (??):
For ANOVA we work with variance (in place of SD used in t-tests)
€
X 1 = 20X 2 = 30
€
X 1 = 20X 2 = 30X 3 = 35
Effect of fertilizer on plant height (cm)
Population of seeds
Randomly select
40 seeds
HEI
GH
T of
pla
nt (C
m)
No Fertilizer
10 gm Fertilizer
20 gm Fertilizer
30 gm Fertilizer
(too much of a good thing)
Why use ANOVA: (a) ANOVA can test for trends in our data. (b) ANOVA is preferable to performing many t-tests on the same data (avoids increasing the risk of Type 1 error).
Suppose we have 3 groups. We will have to compare:
group 1 with group 2 group 1 with group 3 group 2 with group 3
Each time we perform a test there is (small) probability of rejecting the true null hypothesis. These probabilities add up. So we want a single test. Which is ANOVA.
2
(c) ANOVA can be used to compare groups that differ on two, three or more independent variables, and can detect interactions between them.
scor
e (e
rror
s)
Age-differences in the effects of alcohol on motor coordination:
Alcohol dosage (number of drinks)
Independent-Measures ANOVA:
Each subject participates in only one condition in the experiment (which is why it is independent measures).
An independent-measures ANOVA is equivalent to an independent-measures t-test, except that you have more than two groups of subjects.
Logic behind ANOVA: Example
Effects of caffeine on memory:
Four groups - each gets a different amount of caffeine, followed by a memory test. Variation in the set of scores comes from TWO
sources:
• Random variation from the subjects themselves (due to individual variations in motivation, aptitude, mood, ability to understand instructions, etc.)
• Systematic variation produced by the experimental manipulation.
3
Random variation
Systematic variation
+ Random variation
Large value of F: a lot of the overall variation in scores is due to the experimental manipulation, rather than to random variation between subjects.
Small value of F: the variation in scores produced by the experimental manipulation is small, compared to random variation between subjects.
systematic variation random variation (‘error’) F =
ANOVA compares the amount of systematic variation to the amount of random variation, to produce an F-ratio:
Analysis of variance implies analyzing or breaking down variance. We start by breaking down ‘Sum of Squares’ or SS.
sum of squares
€
= (X − X )∑2
We divide SS by the appropriate "degrees of freedom" (usually the number of groups or subjects minus 1) to get variance.
9 calculations
3 3
1 1
1
4
step 1 The null hypothesis:
€
H0 :µ1 = µ2 = µ3 = µ4
steps 2, 3 & 4 Calculate 3 SS values:
1) Total 2) Within Groups 3) Between Groups
No treatment effect α = .05
Total SS
Three types of SS
€
= (Xi −G )2∑
step 2
€
G = 9.5
€
SSTotal = 297
Within groups SS step 3
€
SS1 = (Xi − X 1)2∑
€
SS2 = (Xi − X 2)2∑
€
SS3 = (Xi − X 3)2∑
€
SS4 = (Xi − X 4 )2∑
€
X 1 = 4
€
X 2 = 9
€
X 3 =12
€
X 4 =13
SSwithin groups= 52
Between groups SS
€
X 1
step 4
€
X 2
€
X 3
€
X 4SSbetween groups
€
= n (X 1 −G )2 + (X 2 −G )2 + (X 3 −G )2 + (X 4 −G )2[ ]
SSbetween groups= 245
5
Calculating df (total, within groups and between groups)
dftotal = All scores – 1 = 19
dfwithin groups= df1 + df2 + df3 + df4 = 16
dfbetween groups = Number of groups – 1 = 3
step 5
The ANOVA summary table:
Source: SS df MS F Between groups 245.00 3 81.67 25.13 Within groups 52.00 16 3.25 Total 297.00 19
(Total SS) = (Between-groups SS) + (Within-groups SS)
(total df) = (between-groups df ) + (within-groups df )
step 6 Calculating the Within groups and Between
groups Variance or Mean squares (MS)
Computing F and Assessing its significance:
The bigger the F-ratio, the less likely it is to have arisen merely by chance.
Use the between-groups and within-groups df to find the critical value of F.
Your F is significant if it is equal to or larger than the critical value in the table.
step 7
F = MSbetween groups
MSwithin groups
Here, look up the critical F-value for 3 and 16 df
Columns correspond to between-groups df; rows correspond to within-groups df
Here, go along 3 and down 16: critical F is at the intersection.
Our obtained F = 25.13, is bigger than 3.24; it is therefore significant at
p < .05
6
Interpreting the Results: A significant F-ratio merely tells us is that there is a statistically-significant difference between our experimental conditions; it does not say where the difference comes from.
In our example, it tells us that caffeine dosage does make a difference to memory performance.
BUT suppose the difference is ONLY between: Caffeine VERSUS No-Caffeine
AND There is NO difference between: Large dose of Caffeine VERSUS Small Dose of Caffeine
To pinpoint the source of the difference we can do:
(a) planned comparisons - comparisons between (two) groups which you decide to make in advance of collecting the data.
(b) post hoc tests - comparisons between (two) groups which you decide to make after collecting the data: Many different types - e.g. Newman-Keuls, Scheffé, Bonferroni.
Assumptions underlying ANOVA:
ANOVA is a parametric test (like the t-test)
It assumes:
(a) data are interval or ratio measurements;
(b) conditions show homogeneity of variance;
(c) scores in each condition are roughly normally distributed.
Using SPSS for a one-way independent-measures ANOVA on effects of alcohol on time taken on a motor task.
Three groups:
Group 1: two drinks Group 2: one drink Group 3: no alcohol
7
Data Entry
RUNNING SPSS (Analyze > compare means > One Way ANOVA)
8
Click ‘Options…’ Then Click Boxes: Descriptive; Homogeneity of variance test; Means plot
SPSS output
Trend tests:
(Makes sense only when levels of IV correspond to differing amounts of something - such as caffeine dosage - which can be meaningfully ordered).
Linear trend: Quadratic trend:
(one change in direction)
Cubic trend: (two changes in direction)
With two groups, you can only test for a linear trend. With three groups, you can test for linear and quadratic trends. With four groups, you can test for linear, quadratic and cubic trends.
9
Conclusions:
One-way independent-measures ANOVA enables comparisons between 3 or more groups that represent different levels of one independent variable.
A parametric test, so the data must be interval or ratio scores; be normally distributed; and show homogeneity of variance.
ANOVA avoids increasing the risk of a Type 1 error.
New online course evalua/on ques/onnaires
Please take a few minutes to give us your feedback on your Spring Term Courses.
Opens: February 26th 2010 Closes: March 19th 2010
Access them via your Course Resources page in Sussex Direct.
Completely anonymous.
Your feedback provides invaluable informa/on to guide tutors in teaching and course development.