EPSY6210

02.08.2010

Becoming a Behavioral Science Researcher (another helpful title)

Research Methods in Anthropology (Bernard Russell -- univariates chapter; takes only about a page to explain most concepts)

How regression is like/unlike ANOVA

ANOVA: grouping variable, outcome variable

• (1 or more independent variables, which may have various levels (group 1--farming method 1; group 2, etc.))
• what level is independent variable (predictor) in ANOVA? nominal (categorical, grouping variable)
• the outcome variable is: interval (continuous)
• you can have more than one independent variable in an ANOVA

regression

• continuous outcome (also) -- only one
• its predictors (independent variables) can be categorical or (continuous?)
• if ANOVA can do it, regression can do it

purposes of regression:

• prediction (can this variable predict this other variable?)
  • want a big effect size (a big R squared)
  • don't care "why" necessarily
• explanation (theory testing)
  • also care about "why" (which of the predictors, if any (or all) predict this best)
  • helps explain or inform a theory, supports it
  • understand which predictor contributed to a prediction

example: equation for predicting height at age 20 using height at age 2

• predictor: height at age two  (x)
• outcome variable: height at age 20  (y)
• Y hat = 0 + 2X

another example: predicting 9th grade algebra scores...

• r squared = .60 (predictor can explain 60% of the variance)
• y hat = 3 + 1.5x
• once you have your equation; use it to plug in for the next group of subjects to make predictions for them
• use data (regression equation) to make quality decisions
  • you can be confident in the predictions if you have a large r squared (effect size)

ANOVA Review

• SOS total = the differences (variability) of all the scores on the outcome (dependent variable)
• SOS between =does the mean of each of the groups (grouped independent variables) differ? (null hypothesis)
  • SOS of the means (how different are the groups?) = take the four group means and compute the SOS (grand mean = mean of all scores (the mean of the means, if all groups have the same number of scores)
  • degrees of freedom between = number of groups, minus 1
• SOS within = how different are all the scores in a single group? + the difference of the scores (within) the other groups... (SUM)
  • degrees of freedom within = SUM(group 1's n-1 + group 2's n-1) OR n minus the number of cells (number of unique groups/observations)
• statistical significance says is it likely with a similar population, random sampling that we would come up with the same mean?
• mean of squares SHOULD be called variance = ratio of the sum of squares over the number of observations
• f calculated = MSb/MSw = look in a table to find f critical value (all f distributions are positively skewed because we used squares and cannot have a negative; f calc shows us spreadoutness--you can't have less spreadoutness than none (<0).)
  • if you reject the null hypothesis (that there are no differences), then you declare that there are differences
  • what influences how big or small f calc is? (the bigger, the more statisically significant)  ALSO TRUE FOR REGRESSION
    • sample size (degrees of freedom) -- (more subjects per group = more power)
    • number of groups (more groups = less power)
    • SOS (smaller SOS = less power)
• eta squared = overall effect size from an ANOVA
  • eta squared = SOSb / SOSt
  • how much of the differences in output (coffee output) is due to differences between groups (farming method)? = eta squared
  • for most social science research, effect sizes as large as .6 are rare

Area World (not in the original metric; has a power of 2 ; squared)

• variance
• sum of squares (SOS)
• r squared
• eta squared (n); an effect size from an ANOVA (really the same as r squared)
• COV (covariance)

Score World (in metric of the original scores)

• unstandardized

  • mean

  • standard deviation

  • skewness

  • kurtosis

  • pearson r

  • (raw scores: GPA, SAT, etc.)

• standardized

  • Z score

Case-One Regression (Score World)

• = 1 predictor, 1 outcome (same as Thompson paper)
• line of best fit-- best prediction
• best-case scenario (perfect prediction) -- all dots line up in graph -- r = 1 -- y hat = 1.63 + 5x -- error = y minus y hat = 0 (no error if perfect prediction)
  • the factor times x = slope (slope = rise/run)
  • standardized version: line must go thru centroid; in Z score form, mean of X and Y is now 0 (so line goes thru point of origin, y/x intercept)
    • y hat = 0 + (beta times X)  -- instead of unstandardized (B times X)  --  if you say BETA, then the equation is in STANDARDIZED form
• worst-case scenario -- your prediction is 0 (r = 0); no relationship between predictor variable and outcome variable;  y hat = ? + 0x (which means the line is FLAT, and also goes through the centroid)
  • if you have ZERO INFORMATION to predict, then always predict the MEAN (the most scores fall under the mean on a normal distribution)
  • standardized version: mean of y is 0, so regression line is right along the X axis (y hat = 0x)
• real-world scenario (some prediction; in the middle) -- y hat = .3 + .9x (we're assuming r = .5)
  • standardized version: like the other standardized graphs, the centroid is the point of origin (and the line always goes through the centroid); y hat = beta times x
  • in the real world, what does a slope of .5 mean?  =  if we move one unit on x, y increases by .5
  • when interpreting slope, you must invoke the standard deviations for Y and X (otherwise you have no context)
  • you can also standardize Y and X so that they are in the same scale/context (their standard deviations must be the same)

convert B to Beta:  b = Beta(SDy/SDx)

• b = slope  ( Y hat = a + bX )
• take the standard deviations into account, to standardize the b into a beta score
• Thompson page 7
  • IMPORTANT FOR HOMEWORK

Back to Area World...

• to get from Score World to Area World, we SQUARE it
• thus, square-root to get from Area World to Score World

perfect prediction

• draw a box on the graph for Y -- this is the SOSy (because it's squared) (this is also SOSt of the dependent variable)
• how much of this is y hat (predicted scores), and how much is error?
  • if it's a perfect prediction, all of the SOSy is y hat
• how much of SOSx is useful for predicting SOSy?
  • the SOSx can be larger than SOSy, because there can be more  variability for x than for y
  • BUT the dependent variable drives the bus, so we really care about how much of SOSx is explained by SOSy?
  • SOSx that does NOT overlap is IRRELEVANT
• r squared = 1
• square root = r = plus or minus 1

no prediction

• the square graph for y and the square graph for x can be anywhere, as long as they DO NOT overlap (there is no relationship)
• r squared = 0
• r = 0

real-world prediction (middle)

• some overlap (but not total)
• r squared = .25
• r = plus or minus .5
• 25% of y is explained by x (25% = y hat)
• 75 % is not explained = 75% is error
• SOSy = SOSe + SOSyhat  (SOSe is sum of squares of error)
  • any SOSx that does not overlap with SOSy is irrelevant (useless info)
• how many variables are there in this scenario? (if n = 50) -- FOUR = y, x, e (error), y hat
• synthetic variables = unobserved, unmeasured (created from something else)
  • error and y hat are synthetic, because they are unobserved
• SOSy (total y SOS) = explained (y hat) + error

beta = r(rx)