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Statistics (cont.) Psych 231: Research Methods in Psychology Quiz 10 (chapter 7) is due on Nov. 13th at midnight Journal Summary 2 assignment Due in class NEXT week (Wednesday, Nov. 18th) <- moved due date Group projects Plan to have your analyses done before Thanksgiving break, GAs will be available during lab times to help Poster sessions are last lab sections of the semester (last week of classes), so start thinking about your posters. I will lecture about poster presentations on the Monday before Thanksgiving break. Announcements 2 General kinds of Statistics Descriptive statistics • Used to describe, simplify, & organize data sets • Describing distributions of scores Population Inferential statistics used to generalize back Inferential statistics • Used to test claims about the population, based on data gathered from samples • Takes sampling error into account. Are the results above and beyond what you’d expect by random chance? Statistics Sample Purpose: To make claims about populations based on data collected from samples What’s the big deal? Example Experiment: Group A - gets treatment to improve memory Group B - gets no treatment (control) Population After treatment period test both groups for memory Results: Group A’s average memory score is 80% Group B’s is 76% Is the 4% difference a “real” difference (statistically significant) or is it just sampling error? Inferential Statistics Sample A Treatment X = 80% Sample B No Treatment X = 76% Step 1: State your hypotheses Step 2: Set your decision criteria Null hypothesis (H0) Alternative hypothesis(ses) (HA) Type I error (α): concluding that there is an effect (a difference between groups) when there really isn’t. Type II error (β): concluding that there isn’t an effect, when there really is. Step 3: Collect your data from your sample(s) Step 4: Compute your test statistics Step 5: Make a decision about your null hypothesis Reject H0 Fail to reject H0 “statistically significant differences” “not statistically significant differences” Testing Hypotheses Example Experiment: Group A - gets treatment to improve memory Group B - gets no treatment (control) After treatment period test both groups for memory Results: Group A’s average memory score is 80% Group B’s is 76% Is the 4% difference a “real” difference (statistically significant) or is it just sampling error? Two sample distributions Experimenter’s conclusions XB XA 76% 80% About populations H0: μA = μB H0: there is no difference between Grp A and Grp B Reject H0 Fail to Reject H0 Summary to this point Real world (‘truth’) H0 is correct H0 is wrong Type I error a Type II error b “Statistically significant differences” When you “reject your null hypothesis” • Essentially this means that the observed difference is above what you’d expect by chance • “Chance” is determined by estimating how much sampling error there is • Factors affecting “chance” • Sample size • Population variability Statistical significance Population mean Population Distribution x n=1 Sampling error (Pop mean - sample mean) Sampling error Population mean Population Distribution Sample mean x n=2 x Sampling error (Pop mean - sample mean) Sampling error Generally, as the sample Population mean size increases, the sampling error decreases Sample mean Population Distribution x x n = 10 x x x x x x xx Sampling error (Pop mean - sample mean) Sampling error Typically the narrower the population distribution, the narrower the range of possible samples, and the smaller the “chance” Small population variability Sampling error Large population variability These two factors combine to impact the distribution of sample means. The distribution of sample means is a distribution of all possible sample means of a particular sample size that can be drawn from the population Population Distribution of sample means Samples of size = n XA XB XC XD “chance” Avg. Sampling error Sampling error More info Tests the question: Are there differences between groups due to a treatment? Real world (‘truth’) H0 is correct Reject H0 Experimenter’s conclusions Fail to Reject H0 Two possibilities in the “real world” H0 is true (no treatment effect) One population Two sample distributions XB XA 76% 80% “Generic” statistical test H0 is wrong Type I error Type II error Tests the question: Real world (‘truth’) H0 is correct Are there differences between groups due to a treatment? Reject H0 Experimenter’s conclusions Fail to Reject H0 Two possibilities in the “real world” H0 is true (no treatment effect) H0 is wrong Type I error a Type II error b H0 is false (is a treatment effect) Two populations XB XA XB XA 76% 80% 76% 80% People who get the treatment change, they form a new population (the “treatment population) “Generic” statistical test XA XB ER: Random sampling error ID: Individual differences (if between subjects factor) TR: The effect of a treatment Why might the samples be different? (What is the source of the variability between groups)? “Generic” statistical test XA XB ER: Random sampling error ID: Individual differences (if between subjects factor) TR: The effect of a treatment The generic test statistic - is a ratio of sources of variability Computed Observed difference TR + ID + ER = = test statistic Difference from chance ID + ER “Generic” statistical test The distribution of sample means is a distribution of all possible sample means of a particular sample size that can be drawn from the population Population Distribution of sample means Samples of size = n XA XB XC XD Difference from chance Sampling error Avg. Sampling error Things that affect the computed test statistic Size of the treatment effect • The bigger the effect, the bigger the computed test statistic TR TR + ID + ER + ID + ER ID + ER ID + ER XA XB XA XB Difference expected by chance (sample error) • Sample size • Variability in the population TR + ID + ER ID + ER “Generic” statistical test 1 factor with two groups T-tests • Between groups: 2-independent samples • Within groups: Repeated measures samples (matched, related) 1 factor with more than two groups Analysis of Variance (ANOVA) (either between groups or repeated measures) Multi-factorial Factorial ANOVA Some inferential statistical tests Design 2 separate experimental conditions Degrees of freedom • Based on the size of the sample and the kind of t-test Formula: Observed difference T= X1 - X2 Diff by chance Computation differs for between and within t-tests T-test Based on sample error Reporting your results The observed difference between conditions Kind of t-test Computed T-statistic Degrees of freedom for the test The “p-value” of the test “The mean of the treatment group was 12 points higher than the control group. An independent samples t-test yielded a significant difference, t(24) = 5.67, p < 0.05.” “The mean score of the post-test was 12 points higher than the pre-test. A repeated measures t-test demonstrated that this difference was significant significant, t(12) = 5.67, p < 0.05.” T-test Designs XA XB XC More than two groups • 1 Factor ANOVA, Factorial ANOVA • Both Within and Between Groups Factors Test statistic is an F-ratio Degrees of freedom Several to keep track of The number of them depends on the design Analysis of Variance XA XB XC More than two groups Now we can’t just compute a simple difference score since there are more than one difference So we use variance instead of simply the difference • Variance is essentially an average difference Observed variance F-ratio = Variance from chance Analysis of Variance XA XB XC 1 Factor, with more than two levels Now we can’t just compute a simple difference score since there are more than one difference • A - B, B - C, & A - C 1 factor ANOVA Null hypothesis: XA XB XC H0: all the groups are equal XA = XB = XC Alternative hypotheses HA: not all the groups are equal XA ≠ XB ≠ XC XA = XB ≠ XC 1 factor ANOVA The ANOVA tests this one!! Do further tests to pick between these XA ≠ XB = XC XA = XC ≠ XB Planned contrasts and post-hoc tests: - Further tests used to rule out the different Alternative hypotheses XA ≠ XB ≠ XC Test 1: A ≠ B Test 2: A ≠ C Test 3: B = C XA = XB ≠ XC XA ≠ XB = XC XA = XC ≠ XB 1 factor ANOVA Reporting your results The observed differences Kind of test Computed F-ratio Degrees of freedom for the test The “p-value” of the test Any post-hoc or planned comparison results “The mean score of Group A was 12, Group B was 25, and Group C was 27. A 1-way ANOVA was conducted and the results yielded a significant difference, F(2,25) = 5.67, p < 0.05. Post hoc tests revealed that the differences between groups A and B and A and C were statistically reliable (respectively t(1) = 5.67, p < 0.05 & t(1) = 6.02, p < 0.05). Groups B and C did not differ significantly from one another” 1 factor ANOVA We covered much of this in our experimental design lecture More than one factor Factors may be within or between Overall design may be entirely within, entirely between, or mixed Many F-ratios may be computed An F-ratio is computed to test the main effect of each factor An F-ratio is computed to test each of the potential interactions between the factors Factorial ANOVAs Reporting your results The observed differences • Because there may be a lot of these, may present them in a table instead of directly in the text Kind of design • e.g. “2 x 2 completely between factorial design” Computed F-ratios • May see separate paragraphs for each factor, and for interactions Degrees of freedom for the test • Each F-ratio will have its own set of df’s The “p-value” of the test • May want to just say “all tests were tested with an alpha level of 0.05” Any post-hoc or planned comparison results • Typically only the theoretically interesting comparisons are presented Factorial ANOVAs The following slides are available for a little more concrete review of distribution of sample means discussion. Distribution of sample means Distribution of sample means is a “theoretical” distribution between the sample and population • Mean of a group of scores – Comparison distribution is distribution of means Population Distribution of sample means Sample Distribution of sample means A simple case Population: 2 4 6 8 – All possible samples of size n = 2 Assumption: sampling with replacement Distribution of sample means A simple case Population: 2 4 6 8 – All possible samples of size n = 2 mean mean 2 2 4 6 2 5 2 4 2 6 2 8 4 2 4 4 3 4 5 4 8 6 2 6 4 3 4 6 6 6 8 6 4 5 There are 16 of them mean 8 2 5 8 4 8 6 8 8 6 7 Distribution of sample means 6 7 8 5 4 3 2 1 In long run, the random selection of tiles leads to a predictable pattern 2 3 4 5 6 7 8 means 2 mean 2 2 4 mean 6 5 8 mean 2 5 2 4 3 4 5 4 8 8 4 2 6 6 2 8 6 2 8 6 4 8 8 4 2 3 4 6 6 4 4 6 8 6 4 5 6 7 Distribution of sample means 6 7 8 Shape If population is Normal, then the dist of sample means will be Normal – If the sample size is large (n > 30), regardless of shape of the population Population Distribution of sample means N > 30 Properties of the distribution of sample means • Center The mean of the dist of sample means is equal to the mean of the population Population m Distribution of sample means same numeric value different conceptual values Properties of the distribution of sample means Center The mean of the dist of sample means is equal to the mean of the population Consider our earlier example Population 2 4 6 Distribution of sample means 8 μ= 2+4+6+8 4 =5 5 4 3 2 1 2 3 4 5 6 7 8 means = 2+3+4+5+3+4+5+6+4+5+6+7+5+6+7+8 16 =5 Properties of the distribution of sample means Spread • Standard deviation of the population • Sample size Putting them together we get the standard deviation of the distribution of sample means sX = s n – Commonly called the standard error Properties of the distribution of sample means The standard error is the average amount that you’d expect a sample (of size n) to deviate from the population mean In other words, it is an estimate of the error that you’d expect by chance (or by sampling) Standard error All three of these properties are combined to form the Central Limit Theorem – For any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will approach a normal distribution with a mean of m X and a standard deviation of s n as n approaches infinity (good approximation if n > 30). Central Limit Theorem Keep your distributions straight by taking care with your notation Population σ Distribution of sample means Sample sX μ Distribution of sample means s X Back