## Pages

As a way to knock out a blog post AND do some studying before the final exam, I thought I'd type out - from memory in most cases - some of the concepts and formulas needed for today's test.

Let's start with t-tests, shall we? There are three types of t-tests: simple, independent and dependent. A t-test determines whether the means of two groups are statistically different. For instance, if I find the average number of drinks students have weekly at the beginning of the semester and the average amount of drinks they have weekly at the end of the semester, I could run a t-test to measure whether there is a statistically significant difference.

Simple t-test.
t = ybar 2 - ybar 1 / (s/square root of n)
Degrees of freedom for a simple t-test. df = n-1

Independent t-test.
t = M1 - M2 / Sm
Where Sm = square root of [n1(s1)sq + n2(s2)sq / (n1 +n2) -2 x (1/n1 + 1/n2)]
Degrees of freedom for an independent t-test. df = N-k

Dependent t-test.
t = ED/square root of [n(EDsq) - (ED)sq / n - 1]
Where D = x1 - x2
Degrees of freedom for a dependent t-test. d = N - 1

Once you have a t, you'll look up the "critical value" on a t-table using the degrees of freedom and determine from those numbers (both a positive and negative value) whether your number is significant. Does it fall within the critical value or outside the value? If your number falls outside the tcrit, you'll reject the null hypothesis, basically saying that there IS evidence to show a difference between the means that you were testing.

If you find a statistic that is significant, you'll want to test the effect size. There are two ways of doing that: Cohen's d (1988) and rsq.

d = t[square root of (n1 + n2) / (n1 x n2)]

rsq = tsq / tsq + df

E = sum
sq = squared

ANOVA - this is a process similar to regression. Through ANOVA, you can analyze the variance between and within results. We need to be able to HAND COMPUTE One-way, Two-way and Three-way ANOVA for the exam.

Let's start with a One-way ANOVA. We are going to begin with building an ANOVA table from imaginary data

First, compute the Grand Mean.
GM = n1(m1) + n2(m2) / n1+n2

Next, find the Sum of Squares between, within and total.
SSb = n1(m1-GM)sq + n2(m2 - GM)sq

SSw = (n1 - 1)(s1sq) + (n2 - 1)(s2sq)

SSt = SSb + SSw

Then, find Degrees of Freedom

dfb = k-1
dfw = N-k

Next, compute the standard means

MSb = SSb/dfb
MSw = SSw/dfw

F = MSb/MSw

The F stat is what you can now use to measure whether there is a difference. Figure out the Fcrit using the degrees of freedom.

Fcrit = df = k-1/n-k

Then, go to the F table, find the value and if your number is greater than the Fcrit, you have evidence to show there IS a statistically significant difference.

Don't forget to calculate Eta squared (N2). That measures the effect size just as Cohen's d and rsq measures it for a t-test (see above).

N2 = SSb / SSt

Unfortunately the F stat can't tell you WHERE the difference is so you can utilize a post-hoc test to show where the difference lies.

Those include: Tukey's HSD, Scheffe's test, Dunnett's test and Bonferroni's test.
HSD = q[square root (MSw / n)]

Two-way ANOVA has some crazy degrees of freedom that I need to get back to studying now...