DRIP

To reduce total squared error

(yi), (yhati)

Model Residuals

• - Sum of the measure of model fit 
• - smaller = better
• - Convenient/intuitive measure of error of prediction

dangrump~dansleep

Σ(Yi -Yihat)2

• Σ(Yi - Yibar)2
• bar=mean

RSS/TSS

(TSS-RSS)/TSS

• 1) a set of parameters
• 2) a rule for combining the parameters

• 1) weights
• 2) coefficients
• minimize RSS

RSS, coefficient of determination (R2)

• 1) Specify a null-hyp
• 2) Identify a test-statistic of interest
• 3 Determine the sampling distribution of test stat under assumption that null hyp is true (plus any other assumptions you have to make so it will work)

• 1) Collect data
• 2) calculate value of test stat
• 3) Compare this value against relevant sampling distribution if null hyp is true
• 4) If probability of observing at least this value is smaller than some criterion, reject null hyp

is used to test our null hype under the linear model

• yi= outcome
• b0= coefficient
• b1 = slope
• x = predictor
• e = residual/error

• If 2+ UNCORRELATED predictors, total proportion of explained variance is additive
• Additive models dont include any interaction

If 2+ CORRELATED predictors, total proportion of explained variance is sub-additive

CORRELATION between Y and X with the effect of Z removed from X

CORRELATION between X and Y with the effect of Z removed from both X and Y

• Adding non-predicitve predictors reduces efficacy of model
• But salient predictors remain salient
• Suggests paying more attention to tests of individual coefficients

• Very high correlation btw 2 predictors INFLATES their standard errors
• None of the coefficients of the correlated predictors may be significant
• Suggests paying more attention to the tests of all the coefficients

• Way of checking for (multi) collinearity
• If regress a redundant predictor onto all other predictors, resulting R2 will be very high
• Rule = abandon hope if VIF>10

Start with no predictors and add ones you think will work

Start with all predictors and pull one ones that don't work

• Planned:
• - A priori
• - Confirmatory
• - Hypothesis testing
• - Minimal/controlled capitalisation on chance

• Unplanned:
• - Post hoc
• - Exploratory
• - Prediction
• - Maximal/uncontrolled capitalisation on chance

• Regression model based on theoretical/practical context
• Hypotheses determined by questions of interest
• Mostly avoids capitalisation on chance

• 1. Residuals have no discernible structure - outcome is modelled as a LINEAR function of the predictors (multiply predictors by a coefficient, add together -> "prediction")
• 2. Residuals are independent (uncorrelated)
• 3. Residuals are normally distributed (mean of 0, some kind of SD)
• 4. Residuals have a constant variance (Homoscedasticity)
• 5. No outliers (no residuals distorting results likely to find)

• Quantile Probability Plots
• Shapiro-Wilk test (W closer to 1 = not normally distributed?)

• Ignore it
• Transform 1 or more variables
• Try another more complicated approach

• Population SD is same in both gps
• Chi sq
• P>.05=homosc is violated

ANOVA deals with factors, MULTIPLE REGRESSION deals with numeric

• Compare 2 means of an outcome variable
• The 2 gps are defined by the levels of intervention: (No, Yes)
• Null hyp = both means are the same

• Think of gps as CATEGORICAL variable (or "factor") and conduct t-test accordingly 
• Think of gps as defining a numeric (dummy) predictor: 1 gp has 1 level of predictor, other gp has other level of predictor

• a test of the regression equation: y=b0+b1x+e
• (x is a dummy variable coding for the group, and the null hyp is equivalent to H0:b1=0)

• Only got one variable using to predict outcome, and multiple levels.
• If two levels, its called t-test?

• 2 or more factors are orthogonally (independent of each other) combined/crossed (aka fully crossed design)
• Each CELL defined by choice of level across all factors (factors= Treatment & Expectations)
• Allows effects of multiple factors to be estimated SIMULTANEOUSLY
• Reduces residual error

If each cell has SAME no of observations, its called balanced design

• Effect size
• Partial eta sq same as partial R2??????

• Any departure from additive model = interaction
• Effect of one factor not same at each level of other factor
• Effect of one factor DEPENDS on level of other, effects of the two factors are NOT INDEPENDENT 
• Interaction is an EFFECT: has a size + can be measured

• They are fully crossed
• There are equal number of observations in each cell

• 1. The common variance is assigned to one of the correlated factors or
• 2, The common variance is assigned to non of the correlated factors
• (Anova takes option 1 - may not be appropriate)

• Allocate order in which factors enter the model
• Called "Sequential Sums of Squares/Type I"
• SAME as forward selection
• Method assumed by ANOVA

• Allocate ONLY UNIQUE variance
• Only works if NO interaction

• Do not allocate ANY common variance to any factor/interaction
• Works if SIGNIFICANT interaction
• But main effects may be reduced

3x2 anova

6 (3x2)

the levels of the 2 factors

• F is a ratio of mean squares
• Divide its mean square by the residual

• A planned comparison - (tests meaningful hypothesis)
• A linear combination of predictors that sum to zero
• Another way of specifying dummy variables

• If you don't have any meaningful hypotheses can conduct set of these
• EXPLORATORY rather than confirmatory
• Proposed after meaningful hyps have been tested against data
• Compare mean of each cell with mean of every other cell controlling for the FAMILYWISE ERROR RATE

reject hyp when its true

If you test k hypotheses, probability of making at least 1 type 1 error cannot be less than 1-(1-a)^k

Hybrid form of multiple regression + ANOVA

• 1 or more CATEGORICAL factors (as dummy variables/contrasts) + 1 or more CONTINUOUS predictors (called covariates)
• (Interest usually lies in the effects of the FACTORS on the DV)

• 1) To reduce residual error/variance
• 2) To "control" for possible confounding effects of the covariate(s)

moderately, weakly

The paired samples t-test is based on the variance of the differences between conditions across participants.

This is the same as the interaction between the factor (time) and the subjects variable (id).

• 1. Independence of "subjects" (assume unrelated to each other)
• 2. Normal distribution within each cell
• 3. Sphericity

Assumption that the VARIANCES of differences between each pair of within-subjects cells are EQUAL

• Mauchly test of sphericity
• W=.0004, p<2.2e-16
• = REJECT hypothesis that variances of differences are equal

• These corrections are applied to the degrees of freedom of an F-ratio
• in order to adjust for failure of the sphericity assumption in repeated
• measures anova.

• Expository: explain/provide info
• Productive: generate new info

• Diagrams that exhibit relationship between 2 sets of numbers as a set of points having coordinates determined by the relationship (plots).
• Used to illustrate relationships (charts)

• Aesthetics - maps data onto logical elements of graph
• Geometrics - specifies how elements of graph are represented
• Themes - Modifies look/feel of graph elements
• Others