Univariate Statistics

Factors are single variables which are either observed or manipulated; factors ususally refer to independent variables. A single factor or variable may have several levels (e.g., in the following 2 group t test, the IV or factor of teaching method has 2 levels).

X1 = Grp 1 = Method 1

X2 = Grp 2 = Method 2

Y10

Y11

Y12

Y20

We ususally conceptualize factors as being one of two types: fixed or random.

Fixed factors: For a particular factor, we assume that the levels of the factor that we have represented in our experiment compose all possible levels of the factor in the population. That is, we have presented all possible levels (e.g., types of teaching method) in our experiment.

Random Factors: We assume that the levels of a factor in our experiment have been randomly assigned from all possible levels of the factor that exist in the real world.

How we conceptualize the sampling procedures for the levels of a factor in some study influences how we can generalize our obtained results. With fixed factors, we can only generalize our experimental results only to the levels of the IV we have selected, while results fom random factors can be generalized to the whole population of IV values.

**Note: we almost always consider the subjects used in our experiments to be a random factor. Can you understand why?

We will initially be discusing 2 basic ANOVA designs:

Simple Randomized Design (SRD): Ss are randomly assigned to k experimental groups (or equivalently, to k levels of the IV); or, Ss are nested under each level of the 1V (each experimental group has its own unique Ss group).

S10

S11

S12

S20

Sk1

Sk2

S(k+1)10

Repeated Measures Design (RMD): An extension of the matched t test: each S is tested under several experimental conditions, whereas in the SRD case each S is tested once in the SRD case each S is tested once in one experimental condition. RMD allows repeated observations of Ss across all levels of the IV.

S10

As with all of our previous statistics, we are interested in deriving some index of departure from our null hypohtesis. And, just like before, we need to derive an estimate of the variance which will, in turn, serve as the basis for our statistic. Recall that this is then used in conjunction with a sampling distribution to obtain probabilities and critical values. For the ANOVA, this index of departure is determined by the following concepts.

Mean Square: A mean square (or MS) is some estimate of the variance based on certain sources of variation available to us in our experiment. For example, in the SRD we can estimate the variance one of 3 ways:

from the deviation of subjects' scores from the grand mean (SS total)

from the deviation of teatment group means from the grand mean (SS between)

from the pooled variation of S scores around their respective group means (SS within)

Any mean square is simply some particular sum of squares divided by its appropriate degrees of freedom.

MS total=SS/N-1

MS between=SS/k-1

MS within=SS/N-k

Where N=total # Ss, k=#groups

F-Ratio: The F-ratio is a ratio of 2 mean square values. For the SRD, F is simply equal to MS between/MS within. When Ho is true, MS between and MS within tend to estimate the same value (variance in the DV). That is, under Ho, E(MSbetween) = E(MSwithin) = variance in the DV. However, when Ho is not true, MSbetween becomes a biased estimator of the variances because variance among treatment groups now reflects systematic mean differences among the groups. That is, under H1,

MSbetween/MSwithin = treatment + error variance/error variance

Sampling Distribution of the F-Ratio: Each of the mean square values has its own sampling distribution based on the number of degrees of freedom. More specifically, MSbetween is distributed as chi-square with k- 1 degrees of freedom, while MSwithin is distributed as chi-square with N-K degrees of freedom. Thus, any F ratio has 2 degree of freedom components:

F(k-1, N-k) = MS_between/MS_within