Comparisons Between Means Flashcards
Critical F’s for comparisons use the
degrees of freedom for the numerator and the denominator of the F-ratio.
•There are 1 and 12 degrees of freedom for this comparison.
•Fcritical(1, 12) for p≤0.05=4.75
•Given that Fobservedl=14.29, we can reject the null hypothesis and conclude that A1 leads to better scores than A2
If we consider the ratio of the between groups variability and the within groups variability
Differences among treatment means / difference among subjects treated alike
Then we have:
Experimental error + treatment effects /
Experimental error
Analysis of variance uses the
ratio of two sources of variability to test the null hypothesis
•Between group variability estimates both experimental error and treatment effects
•Within subjects variability estimates experimental error
Between group variability estimates both
experimental error and treatment effects
Within subjects variability estimates
experimental error
In order to evaluate the null hypothesis, it is necessary to
transform the between group and within-group deviations into more useful quantities, namely, variances.
Analysis of variance is the
statistical analysis involving the comparison of variances reflecting different sources of variability
For the purposes of analysis of variance a variance is defined as follows
Variance =
Sum of squared deviations from the mean / degrees of freedom
The sums of squares
From the basic deviations
AS-T= (AS-A) + (A-T)
A similar relationships holds up for the sum of squares
In other words
SStotal = SSwithin + SSbetween
A basic ratio is defined as
(Score or sum) to the power of 2 /
Divisor
The numerator term for any basic ratio involves two steps:
- the initial squaring of a set of quantities
* summing the squared quantities if more than one is present.
The denominator term for each ratio is the
number of items that contribute to the sum or score.
Basic Ratios make the
calculation of the sum of squares relatively simple
distinctive symbol to designate basic ratios and to distinguish among them
AS = the basic observations or scores
A = the treatment sums
T= the grand sums
The sums of squares can be calculated by combining these basic ratios:
Total Sum of Squares
AS-T
Between Group Sum of Squares
A-T
Within Group Sum of Squares
AS-A
The ratio we are interested in is the
ratio of the between groups variability and the within groups variability
Variability is defined by the equation:
SS / df
•where SS refers to the component sums of squares and df represent the degrees of freedom associated with the SS
The degrees of freedom associated with a sum of squares correspond to the
number of scores with independent information which enter into the calculation of the sum of squares
Degrees of freedom are the number of
observations that are free to vary when we know something about those observations
The F-Ratio is defined as
F = MSA / MSSA
The results of the ANOVA are usually displayed by
Computer programs in a summary table
In order to decide whether or not the null hypothesis is rejected we need to
find out what value of F is necessary to reject the null hypothesis
- There is a simple rule for this.
- Reject H0 when Fobserved> Fcritical otherwise do not reject H0
- We obtain a value for Fcritical by looking it up in the F tables.
To find the Critical value
- Take the degrees of freedom for the effect (A) and look along the horizontal axis of the F table.
- Take the degrees of freedom for the error term (S/A) and look down the vertical axis of the F table.
Where is the Critical value of F?
The place were the column for the degrees of freedom of the effect A meets the row for the degrees of freedom of the error (S/A)
The F-ratio includes information about…
all the levels of the independent variable that we have manipulated
What is the Omnibus F Ratio?
The F-ratio for an overall difference between the means as reported in the ANOVA summary table
The best the Omnibus F-ratio can tell us
- is that there are differences between the means.
* It cannot tell us that what those differences are.
With a nonsignificant omnibus F we are prepared to
assert that there are no observed differences among the means
We can stop the analysis here
A significant omnibus F…
- demands further analysis of the data.
* Which differences between the means are real and which are not?
Analytical comparisons
- Before we set out to collect the data, we could have made specific predictions about the direction of the effects
- In this case we can use a technique known as planned (a priori) comparisons.
- We might have designed an experiment where we couldn’t be precise enough to say what the differences would be.
- In this case we use post hoc (after the event) comparisons.
Before we set out to collect the data…
we could have made specific predictions about the direction of the effects
Planned (a priori) comparisons
made specific predictions about the direction of the effects before collecting data
We might have designed an experiment where we couldn’t be precise enough to say what the differences would be…
In this case we use post hoc (after the event) comparisons.
Planned comparisons are based on the calculation of an…
f ratio
- This requires us to calculate the variability inherent in the comparison
- We calculate a sum of squares associated with the comparison. This is given by:
N (psychology sign) to the power of 2 / c to the power of 2
Psychology sign- difference between the compared means
N- number of subjects that contribute to the mean
C- the coefficient with which we weight the mean
A F ratio is calculated to test the comparison. For this a …
Mean square is required
