Test 2

r family

"variance accounted for" measures

covariance

# that reflects the extent to which 2 variable covary with one another aka S(hat)xy

calculating n for independent samples t

(2*(noncentrality^2))/(d^2)

Calculating N for one-sample t

(noncentrality/d)^2

Kendall's Tau

*Dr. Woodard's preference, can calculate significance X is ordinal, Y is ordinal or higher Step1: put scores on X in order from lowest to highest P is # of consistent score (if 1st # < 2nd #) Q is # of inconsistent score (if 1st > 2nd #) Tau = P - Q / P + Q

Spearman's rho

*special case of pearson's r X is ordinal, Y is ordinal or higher

ANOVA tree

--->SSbg (variation due to treatment (df=k-1) CF --> SStotal(total variation (df = N-1) --->SSwg (variation due to error(df = N-k)

planned comparisons

-based on theory -do not need significant omnibus F -limited to k-1 comparisons -don't need to control αFW -must be orthogonal

Orthogonal contrasts

-contrasts are NOT correlated -one contrast gives us no information about another contrast

post-hoc comparisons

-not based on theory, exploratory -need significant F -not limited to k-1 -should control αFW

cohen's d rule of thumb

0.2 = small, 0.5 = medium, 0.8 = large

Fcontrast calculation

= (SScontrast1/dfcontrast1)/MSwg SScontrast = ((∑(𝑐𝑖𝑇𝑖)))^2/(𝑛∑((𝑐𝑖)^2 ) 𝑐𝑖 is the number you assigned to the contrast (eg. 1, -1, 0) 𝑇𝑖 is the sum of all the x's in that group

harmonic mean (effective sample size) for unequal sample sizes

= 2(n1)(n2)/n1+n2

eta squared

= SSbg / SStotal simpliest effect size for ANOVA, aka correlation ratio -positively biased -represents proportion of the DV accounted for by group membership

SStotal computation

= Sum(x^2)/1 - CF df= N-1

Correction Factor

= T^2/N

ANOVA

Analysis of Variance

ANOVA via multiple regression

Basic approach: through coding, we break down each level of IV in ANOVA to become each predictor in MR. -In this case, we call each predictor a "vector" because it is no longer a stand alone variable.

Tukey's HSD

Basic rationale: obtain difference between any pair of group means, and compare it to critical difference. compares all pairwise comparisons -controls αFW -2nd most stringent -3rd most powerful

Dunn-Bonferroni

Chosen PW OR PW&OW (make choice a priori) -controls αFW -3rd stringent -2nd powerful

Coding: Orthogonal/contrast

Coding scheme: like orthogonal coding for planned comparisons in ANOVA Constant: grand mean of all levels of IV Slope: use to calculate mean for each level or differences among levels.

correlation

Correlation represents how much two variables (X&Y) covary together in a linear fashion.

Using R2 to Calculate F

For Omnibus F: 𝐹(𝑝,𝑁−𝑝−1)= (𝑅^2⁄𝑝)/(((1−𝑅^2))⁄((𝑁−𝑝−1))) For each comparison: 𝐹(1,𝑁−𝑝−1)=(𝑟𝑐𝑜𝑚𝑝𝑎𝑟𝑖𝑠𝑜𝑛^2)/(((1−𝑅^2))⁄((𝑁−𝑝−1))) p = k-1

One-way ANOVA

IV = 1 nominal, with 2 or more levels DV = 1 interval/ratio

Standard Error of Estimate/prediction

On average, the predicted Y scores deviate from actual Y scores by this much unit =SQRT(SSresidual/N-2)

Dunnett

PW only against control group

Coefficient of Determination for Regression

R^2 = SSregression/SStotal Amount of variance in Y accounted for by X

multiple regression

Regression using multiple predictors and one criterion variable. equation = Y' = b0 + b1X1 + b2X2 + .... -Adding more predictors increases overall variance explained (R2) in Y b0: constant other b/β weights or slopes: unique contribution of each X (without redundantly taking overlap between Xs into account) If Xs are correlated, standardized regression weights (β) are no longer bivariate relationship (r) between X and Y. Predictors/IVs: usually interval/ratio, can be dichotomous Criterion/DV: interval/ratio, continuous

SSwithin

SStotal - SS between DF = dftotal - dfbetween

ANOVA matrix

Set up as follows: x1 x1^2 x2 x2^2 x3 x3^2 sum each column separately Sum of non-square totals = T

Coding: Effect

Similar to dummy coding except for the last group The last level gets all -1 Constant: grand mean of all levels of IV Slope: difference between each level and the grand mean

SSbetween

Sum(Total per group/ n) - CF df = k-1

power

The probability of correctly rejecting a false null hypothesis

One-sample t-test for correlation

Used for testing r to 0 One-sample t-test for correlation t(N-2) = r*SQRT(n-2)/SQRT(1-r^2)

familywise error rate

When k>2, we are making a set/family of comparisons Type I error (α) for every comparison will accumulate/inflate αFW = 1- (1-αPC)^c c = #of comparisons aPC = alpha per comparison

Linear Additive Model for an Individual Score (X)

X = Meu (population mean) + Tau (treatment effect) + E (Error)

General Linear Model

Y = b0 + bX + e ANOVA is a special case of (multiple) linear regression, which itself is a special case of the General Linear Model (GLM) GLM for MR: Y = b0 + b1X1 + b2X2 + ... + e GLM for ANOVA: Yij = μ + τj + eij

Regression equation

Y' = a + bX X: predictor (variable) Y: criterion (variable) Y': predicted value of Y a: intercept (constant) b: slope (constant)

Regression Formula with Z scores

Zypredicted = (rxy)(Zx) rxy= slope intercept = 0

effect size

a simple way of quantifying the difference between two groups that has many advantages over the use of tests of statistical significance alone

biserial

artificial dichotomous x, continuous y

tetrachoric

artificially dichotomous x and y

pearson's r

both x and y are interval/ratio & normally distributed = covxy/sxsy

Correlation Z-score formula

bottom formula used for understanding how z-scores of X & Y affect the correlation.

slope

change in predicted y for a one-unit change in x Sum(x-xbar*y-ybar)/variance of x

fixed effects

chose IVs systematically, generalize only to those chosen IV levels -can also be when all possible levels used (e.g. male/female for sex)

Scheffe

compares all pairwise and otherwise -controls αFW -1st most stringent -4th powerful

Fisher's LSD

compares all pairwise comparisons -doesn't control αFW -least stringent -most powerful

coefficient of determination for correlation

correlation coefficient squared, effect size

noncentrality parameter for one-sample t

d*SQRT(N)

noncentrality for independent samples t

d*SQRT(n/2)

SSregression/SSpredictor

degree of improvement over just using the mean score as a predictor Sum(Ypredicted - Ymean)^2

Omega squared

effect size for ANOVA should be 0 when there is no treatment effect, between 0 and 1 when there is -proportional amount of population variance that is attributed to variance among experimental treatments -the least bias effect size for ANOVA -not good to use when levels of IV are extreme

estimated epsilon squared

effect size for ANOVA, better estimate than eta squared =SSbg - (dfbg)(MSwg)/SStotal aka adjusted R-squared

Cohen's f

extensions of cohen's d, standard deviation among group means/standard error of estimate measure of effect size for ANOVA Small: f = .1 Medium: f = .25 Large: f = .4

Coding: Dummy

for each vector, only one level gets 1, all the other levels get 0. Each level takes turns to get the 1 until one level is left. The last level gets 0s on all vectors. Constant: mean of the level which gets all 0 Slope: difference between each level and the last level

noncentrality parameter

helps define your noncentral distribution and represents the degree to which the mean of the sampling distribution of the test statistic departs from its mean when the null is true Noncentrality Parameter when power =0.8 is 2.8!

Assumptions for ANOVA

independence of error scores normality of error scores homogeneity of variance

point-biserial

kind of pearson correlation x is dichotomous, y is continuous

phi

kind of pearson correlation x and y both dichotomous

Glass's Delta

measure of effect size, modified Cohen's d by replacing the common SD with SD of control group (can't use if no control group)

SSresidual

minimized squared error Sum(Y-Ypredicted)^2

Kruskal-Wallis Test

non-parametric test for ANOVA -each sample is independent -is group has at least 5 participants -uses ranks -good to use when DV is ordinal

Trend Analysis

patterns of means (quantitative IV levels) purpose is to establish FUNCTIONAL relationship IV and DV -used for non-linear trends -weights taken from table of orthogonal polynomials -can apply weights to SScontrast formula SScontrast = ((∑(𝑐𝑖𝑇𝑖)))^2/(𝑛∑((𝑐𝑖)^2 ) -used to obtain F

random effects

randomly chosen levels generalizable beyond chosen IV levels

ANOVA tree for planned comparisons

see notes, too complicated for flashcard

Beta

standardized regression coefficient (aka standardized slope) symbolic representation of the slope Independent of M and SD of X & Y can directly compare to Beta's of other distributions =rxy ONLY IN BIVARIATE REGRESSION

d family

standardized units of difference, the difference between two populations divided by the standard deviation of either population

b

symbolic representation of slope raw unstandardized units affected by M and SD of X & Y not directly comparable

SStotal

total variation in the criterion =Sum(Y-Ymean)^2 OR =SSresidual + SSregression

Confidence interval for an r (to z)

transform into a z CI = Zr +/- Zcrit(SQRT(1/n-3))

testing two correlations - z test

use fisher's r to z transformation Z= Z1 -Z2/SQRT((1/n1-3) - (1/n2-3))

Fisher's r-z transformation

used to see if there is a significant difference between two correlations or for conducting a confidence interval around a correlation

F ratio

variance due to treatment/variance due to error AKA MSbetween groups/MSwithin groups AKA (SSbg/dfbn)/(SSwithin/dfwithin)

Modified Bonferroni Procedure

when you do more than k-1 planned comparison as backed up by theory αPC = αFW/c = (k-1)α/c

Ordinary Least Squares Regression

with the goal of minimizing the sum of the squares of the differences between the observed responses in the given dataset and those predicted by a linear function

one-sample z-test for correlation

z= Zr - Zp/SQRT(1/n-3)

Bonferroni correction for α per comparison

αPC = α/c