Weeks 15 and 16 Flashcards
The independent variable in an ANOVA must be ordinal in nature.
a) True
b) False
b) False
Explanation … In ANOVA, the independent variable should be nominal and without any sense of order or interval. If it does happen to carry some sense of order (for example when the participants in a design are treated with low, medium, or high doses of some drug) then the ANOVA will simply ignore the extra structure.
The F-ratio used in ANOVA cannot have values smaller than 1.
a) True
b) False
b) False
Explanation … The F-distribution used in ANOVA shows the probability of different F-ratio values you would expect to see if the null hypothesis `is true. On the whole, in these circumstances, if the null hypothesis is true then you would expect to see F = 1. But, of course, samples are never perfect reflections of the populations from which they are drawn, and so there is a probability that just by chance when you calculate F after an experiment it might be larger than 1 and also some probability that just by chance it could end up smaller than 1. So it is perfectly possible to have F-values smaller than 1
The F-ratio used in ANOVA cannot have values smaller than 0.
a) True
b) False
a) True
Explanation … The values in the top and bottom of any F-ratio are based on sums of squares and degrees of freedom. Since squared values are always positive, any sum of squares must be positive as well. Furthermore, degrees of freedom are always positive. Since all these elements of F-ratios are positive numbers, the F-ratio itself must be positive.
Suppose that a 1-way ANOVA design has n = 25. If the null hypothesis is true and the value of the variance of sample means happens to be s_M^2=49, what is a good estimate for the population variance?
a) 25 x 49 b) 7 x 25 c) 49 / 25 d) 7 / 5 e) Can’t say based on the information given
a) 25 x 49
Explanation … ANOVA tests use F-ratios where the numerator of the ratio is a between estimate of population variance and the denominator is a within estimate of population variance. If the null hypothesis is true, as we are told it is in this question, then the between estimate of population variance is a legitimate one and depends on the variance of the sample means in the design. The between estimate of population variance in a 1-way design is s_X^2= ns_M^2. Since we are told that n = 25 and ns_M^2=49, we must have s_X^2=25×49.
As in the previous question, suppose that a 1-way ANOVA design has n = 25. If the hypothesis is false and the value of the variance of sample means happens to be s_M^2=49, what is a good estimate for the population variance?
a) 25 x 49 b) 7 x 25 c) 49 / 25 d) 7 / 5 e) Can’t say based on the information given
e) Can’t say based on the information given.
Explanation … if the null hypothesis is false then the between estimate of population variance doesn’t work. This is because we can no longer think of the samples as all being drawn from the same population which, in turn, tells us that the sample means (the Mi) are not members of a sampling distribution for that population. But if they are not members of a sampling distribution then we can’t estimate the standard error of the mean from them or form an estimate of population variance from them. This all sounds like a disaster but it is what ANOVA depends on. If the null hypothesis is false, then the “between” estimate of population variance is no longer any good and it won’t match the (still legitimate) “within” estimate of population variance. That, in turn, means the F-ratio will not equal 1 and this is what we look for to tip us off that the null hypothesis is false.
6) Which of the following is the best description of an “F” distribution?
a) the distribution of possible F-values beyond which you would reject the null hypothesis
b) the ratio of 2 sums of squares
c) the distribution of possible F-values when the treatment is effective.
d) the distribution of possible F-values ratio when the treatment is useless.
d) the distribution of possible F-values ratio when the treatment is useless.
Explanation … In an ANOVA, after calculating an F-value from observations, you need to turn it into a p-value. In any hypothesis test, the p-value you calculate is the probability of getting the result you see experimentally (or a result even more extreme) when the hypothesis you are testing is true. What is needed here, therefore, is the distribution of possible F-values that you expect to see if the null hypothesis of ANOVA is true. Since the null hypothesis can be stated as saying that differences in treatment are useless at controlling the dependent variable, the correct answer for the question is “d)”.
In a 1-way ANOVA, which part of the structural model is SSbetween derived from?
a) deviations between group means and the grand mean
b) deviations between individual observations and the grand mean
c) deviations between individual observations and their group means
d) the overall or “grand” mean
a) deviations between group means and the grand mean
Explanation … SSbetween is a sum of squared terms from the structural model. The 1-way structural model divides each and every observation in a dataset into a “between groups” part that describes how group means deviate from the grand mean, and a “within groups” part that keeps track of how each observation deviates from tits group mean. This division can also be seen as separating the influences of different levels of treatment (the between-groups part) from the effects of unexplained errors that have nothing to do treatments (the within-groups part). SSbetween is just a sum of squared between-groups contributions (one from each observation) and so the correct answer is “a)”.
In a 1-way ANOVA, which part of the structural model is SSbetween derived from?
a) the Tj terms
b) the Xji – u terms
c) the Xji – uj terms
d) u
a) the Tj terms
Explanation … This is exactly the same question as the previous one and has the same selection of answers too (in the same order). The only difference is that the answers here use symbols rather than words. You need to be able to go back and forth between explanations in words and explanations in symbols. As explained in the previous question, SSbetween is based on deviations of group means from the grand mean. In symbols, one way to write a component like this is “j – “, and this would be a legitimate answer to this question, but since we are talking here about characterizing the effects (on the dependent variable) of belonging to a particular treatment group, we often call these the “treatment effects” and label them j. as shown in answer “a)”.
In a 1-way design with k = 4 treatment levels and n = 10 observations per treatment, how many components need to be squared and summed together to find SSError?
a) 4
b) 10
c) 39
d) 40
d) 40
Explanation … A structural model splits every single observation in a design into component pieces. Sums of squares are then constructed by collecting one of components from every observation then squaring them and summing them all together. In particular, the structural model can be used to identify an “error” component for each observation. SSError is constructed by collecting these error components together, squaring each one, and then summing them all.
The question is about a 1-way design with 4 levels and 10 observations at each level. That means there are N = 40 observations. Each observation will have an associated error and so there will be 40 error components overall. SSError is thus a sum of 40 squared components.
In a 1-way design with k = 3 levels and n = 30 observations per treatment, how many components need to be squared and summed together to find SSBetween?
a) 3
b) 30
c) 90
d) none of the above
c) 90
Explanation … This is basically the same as the last question. Whenever you construct a sum-of-squares you always go to each observation, gather an appropriate component piece from it, then square each component and sum them all together. Here you would be gathering “between” components from each one of N = 90 observations. That makes for 90 components that you would then square and sum to get SSBetween.
Suppose that a 1-way design has group means M1 = 1, M2 = 2, and M3 = 3. Also assume that the grand mean is the average of the group means, i.e., M = 2. What is the value of 1?
a) -1
b) 0
c) +2
d) +6
a) -1
Explanation … In a 1-way ANOVA, the components, Tj, that appear in the structural model are called “treatment effects” and are meant to characterize the effect of the treatments on the dependent variable. All participants in a design who are treated in the same way are in the same “treatment group” and therefore all share the same Tj. For a “centred” dataset, where all observations are expressed as deviations from the grand mean, any treatment effect will be expressed as the deviation of a local treatment mean from the grand mean. The question asks about T1 which describes the effect of being in the first group with treatment mean M1 = 1. Since the value of the grand mean is M = 2, we have T1 = M1 – M = 1 – 2 = -1.
In a 1-way ANOVA with 8 treatment groups and 10 participants per group, what is dfwithin?
a) 7
b) 9
c) 72
d) 79
c) 72
Explanation … In a 1-way ANOVA, the within-groups degrees of freedom is the sum of the degrees of freedom found within each treatment group. In this question, each group has 10 participants and so there are 9 degrees of freedom found in each group. Since there are 8 groups, there are 9 8 = 72 within-groups degrees of freedom overall. This reasoning can be turned into the formula – dfwithin = k(n-1) – because the degrees of freedom within each group (in a balanced design) is n-1 and the number of these groups is k.
A completely different way of working out this question is available if you realize that degrees of freedom in a design are partitioned according to the formula dftotal = dfbetween + dfwithin. This tells us that if we know that dftotal = 79 (which is straightforward to calculate since there are 80 participants), and realize that 7 of these total degrees-of-freedom have been used up by the differences between the 8 groups, then the residual must be dfwithin. We thus have dfwithin = dftotal – dfbetween = 79 – 7 = 72.
7) Suppose we have a 1-way ANOVA with k = 8 and n = 10. If SStotal = 366 and SSbetween = 72, what is SSwithin?
a) 80
b) 294
c) 365
d) 438
Explanation …. In solving this problem you should first recognize what it is not about. This question asks for a sum of squares. But SS values do not depend on experimental design, they depend only on the results of experiment. Thus, the information about the experimental design (k = 8 and n = 10) has nothing to do with the answer here.
You can solve this question solely by realizing that in an analysis like this, SStotal is completely partitioned into its 2 parts. Thus, SStotal = SSbetween + SSwithin. Since we know the values of 2 of the SS terms so we can write 366 = 72 + SSwithin. From this it is obvious that SSwithin = 294.
