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Statistics and Research methods
Wiskunde voor HMIBijeenkomst 3Relating statistics and experimental design
Contents
Multiple regression Inferential statistics Basic research designs Hypothesis testing
– Learn to select the appropriate statistical test in a particular research design
Multiple Regression
Multiple correlation– The association between a criterion variable and
two or more predictor variables
Multiple regression– Making predictions with two or more predictor
variables
Multiple Regression
Multiple regression prediction models– Each predictor variable has its own regression coefficient– e.g., Z-score multiple regression formula with three predictor
variables:
Standardized regression coefficients
))(())(())((ˆ3321 21 XXXY ZZZZ ))(())(())((ˆ
3321 21 XXXY ZZZZ
Note: the betas are not the same as the correlation coefficients for each predictor variable (because predictors “overlap”)
Standardized regression coefficient (Beta) of a variable: about unique, distinctive contribution of that variable (overlap excluded)
There is also a corresponding raw score prediction formula for multiple regression:
Ŷ = a + (b1)(X1) + (b2)(X2) + (b3)(X3)
Multiple Regression
Multiple correlation coefficient
R In SPSS output: Multiple RR is usually smaller than the sum of individual
correlation coefficients in bivariate regressionR2 is proportionate reduction in error =
proportion of variance accounted for
Inferential Statistics
Make decisions about populations based on information in samples (as opposed to descriptive statistics, which summarize the attributes of known data)
Notations in statistical test theory
Population Parameter Sample Statistic Basis Scores of entire population Scores of sample only Specificity Usually unknown Computed from data
Symbols Mean M
Standard Deviation SD
Variance 2 SD2
Sample and population
The Normal Distribution (Z-scores)
Normal curve and percentage of scores between the mean and 1 and 2 standard deviations from the mean
Basic research methods
Experimental method– manipulation of variables and measure effects
Field studies – observation
– No outside intervention, e.g. ethnography
Quasi-experimental method– Combination of elements of other two
We concentrate on experiments and quasi-experiments
Experimental method
Manipulation of (levels of) one or more independent variables (e.g. medication: pill or placebo; different versions of a user interface)
experimental conditions Control (keep constant) other possibly intervening
variables Measure dependent variables (e.g. effectiveness,
performance, satisfaction) Test for differences between the conditions
Experimental design
How to assign subjects to conditions?
Between-subjects design– a subject is assigned to only one of the conditions
Within-subjects design orRepeated measures design– Each subjects receives all the experimental
conditions
Between-subjects design
Randomization: assign subject at random to different conditions
Matching: random assignment but control for variable that is expected to be very relevant Example: (if sex is important) seperately
assign men to experimental groupsassign women to experimental groups
Equal amount of men and women in conditions.“the subjetcs in each condition were matched on sex”
Between-subjects design (continued)
Matched pairs – Two subjects that are similar (on relevant variable(s))
assigned to different conditions Randomized blocks design
– Extension of matched pairs for more than two conditions, e.g. 3 conditions
– Form blocks of 3 similar subjects– Assign subjects in one block randomly to different
conditions
Between-subjects design (continued)
Factorial designs – More than one independent variable– Study separate effects of each variable (main effects)
but also interaction between variables– Interaction effect: the impact of one variable depends
on the level of the other variable – Two-way factorial research design (two independent
variables); three-way with three indep. variables – 2x2 if independent variables have two levels
(condions) or 3x3 with three levels
Within-subjects design
Same subjects in each experimental condition Repeated measures design
– Within-subjects design required if change is measured as a consequence of an experimental treatment (e.g. testscores before and after a training)
In other situations: carryover effects– experimental conditions need to be counterbalanced– One half sequence AB the other half BA
Hypotheses Testing
H0: Null hypothesis – No difference– The Independent variable has no effect
e.g. pill or placebo make no difference
H1 (or Ha): Alternative hypothesis – Significant difference– The Independent variable has an effect
Hypothesis Testing Errors
Type I Error:– Null hypothesis is rejected but true.
– Alpha (α) probability of making type I error
Type II Error:– Null hypothesis is not rejected but false.
– Beta (β) probability of making type II error
No effect, but you say there is.
Real effect, but you say there’s not.
Type I and II errors
α usually 0.05 or 0.01
β usually 0.20
H0 Is True H0 Is False Reject H0 Type I error
Right decision
Retain H0 Right decision
Type II Error
Statistical Power
Power:
The probability that a test will correctly reject a false null hypothesis (1- β )
An Example of Hypothesis Testing
A person claims to be able to identify people of above-average intelligence (IQ) with her eyes closed
We devise a test – take her to a stadium full of randomly selected people from the population and ask her to pick someone with her eyes closed who is of above average IQ.
If she does, we’ll be convinced. But she might pick someone with an above-average IQ just by chance.
Distribution of IQ scores is normal with M = 100 and SD = 15 IQ Score Z Score p 145 +3 .13% 130 +2 2% 115 +1 16% So we set in advance a score by which we will be convinced. %chance Z score IQ 2% +2 130 1% +2.33 135 5% +1.64 124.6
The Hypothesis Testing Process
1. Restate the question as a research hypothesis and a null hypothesis about the populations Population 1 Population 2 Research hypothesis or alternative hypothesis Null hypothesis
The Hypothesis Testing Process
2. Determine the characteristics of the comparison distribution Comparison distribution: distribution of the sort you
would have if the null-hypothesis were true.
The Hypothesis Testing Process
3. Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected Cutoff sample score Conventional levels of significance:
p < .05, p < .01
The Hypothesis Testing Process
4. Determine your sample’s score on the comparison distribution
5. Decide whether to reject the null hypothesis
One-Tailed and Two-Tailed Hypothesis Tests
Directional hypotheses– One-tailed test
Nondirectional hypotheses– Two-tailed test