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40 Cards in this Set
- Front
- Back
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When would you use ANOVA? |
If you want to know if there is a statistically significant difference between more than 2 sample means ANOVA does not pinpoint where the difference is; it will just that you if the means differ from each other ANOVA makes an inference that your populations are different |
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Assumptions of ANOVA |
Homogeneity of variance: each of our populations has the same variance Normality: the values for each condition are normally distributed around their mean Independence of observations: the assumption that within a treatment the values are all different from each other (why random assignment of subjects to groups is important) |
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What are the null and alternate hypotheses for ANOVA? |
Null: The population mean's are equal. m1=m2=m3 Alternate: At least one of the means differs from at least one other mean. If you reject the null, it is written as: A one-way analysis of variance revealed that there were significant differences among the means of n groups (F(DFnum, DFden) = x, p = .001) |
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What is F? |
The statistic for ANOVA F = (treatment effect + chance variation)/chance variation As the numerator grows, F grows If there is an effect, you expect F > 1 As effect size grows, F > 1 If F is close to 1 then there is no effect F = MStreat/MSerror If F is < than alpha, reject the null. A significant F is only an indication that not all the population means are equal. |
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Mean square error (MSerror) |
Average of the variances. Estimate of the population variances. Always valid, does not matter if the null is true or false. |
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The logic of ANOVA in terms of estimating common population variance (CPV) |
We want to know if more than 2 populations have statistically different means. We can calculate this by using CPV because our populations should have common population variance. There are two ways to estimate CPV, or population variance: mean square error (MSerror) or mean square treatment (MStreat). MSerror is the average of the variances and is always valid, it does not matter if the null is true or false. MStreat can only be used if the null is true. If the two estimates agree, we fail to reject the Null. If they disagree, we reject the Null and conclude that underlying treatment differences must have contributed to our second estimate, inflating it and causing it to differ from the first. |
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Expected value of a statistic |
Also referred to as the expected MSerror The average value that statistic would assume over an infinite # of repeated sampling |
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Error variance |
The variance unrelated to any treatment differences (variability of scores within the same condition) |
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Summary table |
Summarizes a series of calculations, making it possible to tell at a glance what the data have to offer. We care about the factor (between), error (within), and corrected total. Ignore corrected model, intercept, and total. |
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Levene's test |
Tests the assumption of homogeneity of variance.
It is essentially a t test on the absolute deviations of observations from their sample mean. Does not need to be used if you have equal n's. If significant, you have violated homogeneity of variance. You can correct the violation/heterogeneity of variance using Welch's statistic, which is your new k. |
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5 ways to transform your data |
1) Log transform 2) Square root transform 3) Reciprocal transform 4) Arcsine transform 5) Trimming/Winsorizing |
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When to use log transform |
-Positively skewed data -The standard deviation is proportional to the mean (e.g. as the mean increases the standard deviation increases) |
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When to use square root transform |
-Mean is proportional to the variance -Positively skewed data |
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When to use reciprocal transform |
Very large values on the positive tail |
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When to use arcsine transform |
Proportions |
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Winsorized data |
Use when you have a heavy-tailed distribution, a relatively flat distribution that has an unusual number of observations in the tails Trimmed sample: a sample from which a fixed percentage of the extreme values in each tail have been removed You replace the trimmed values with the most extreme values remaining in each tail Lose a degree of freedom for each value you replace |
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Fixed versus random models of ANOVA |
Fixed: the factor has fixed levels (there is no population, your variables are the only you care about) Random: the factor was pulled from a population |
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Balanced designs |
Experiments designed with the idea of collecting the same number of observations in each treatment. |
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Missing at random |
Referring to data, if people just drop out randomly or classes that you are studying have more or fewer students than others. |
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Why transform your data |
Use transformation when you have markedly skewed data or heterogeneous variances. It is an alternative to Welch's procedure. The transformed data yields homogeneous variances, which you can then run a standard analysis of variance on. |
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Fixed-model ANOVA |
The variable/treatment has fixed levels. The treatment levels are deliberately selected and would remain constant from one replication to another. |
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Random-model ANOVA |
The treatment levels are pulled from a population by a random process, vary, and would be expected to vary across replications. |
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What does effect size tell you? |
An idea of the magnitude of the effect. |
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Eta squared |
Calculation for effect size in an ANOVA. It is biased because it overestimates the effect size in a population. Small = .01 Medium = .06 Large = .15 |
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Omega squared |
Use if you want a better estimate of what the effect size is in the population. Use in a one-way between subjects ANOVA. |
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ANOVA in regression and experiments |
ANOVA in regression: used to assess whether the regression model is good at predicting an outcome . ANOVA in experiments: used to see whether experimental manipulations lead to differences in performance on an outcome (DV). |
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Theory of ANOVA |
We calculate how much variability there is between scores; we then calculate how much of this variability can be explained by the model we fit to the data (Model Sum of Squares) and how much cannot be explained (Residual Sum of Squares). |
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R-based measures |
Eta squared. Percent variance accounted for proportional reduction in error. R squared: how much you have reduced error. |
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d-based measures |
Mean differences. |
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Why use follow-up tests in ANOVA? |
A significant F is only an indication that not all the population means are equal. We need additional tests to find out where the group differences lie. These are called Multiple Comparison procedures. |
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What do we do when we figure out there is a difference? |
Use Multiple Comparison procedures. There is a question of the probability of Type I errors, which is why we specify error rates (the probability of Type I errors). Calculate error rates by using Error Rate Per Comparison or Familywise Error Rate. |
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Error Rate Per Comparison (PC) and Familywise Error Rate (FW) |
PC: The probability of a Type I error on any given comparison. FW: The probability that your family of conclusions contain at least 1 Type I error. |
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A priori and post hoc comparisons (in ANOVA) |
A priori: Choose mean comparisons before the data are collected (also refereed to as "contrasts"). Post hoc: Comparisons are planned after the data has been collected. |
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Comparisons and multiple t-tests |
Use individual t tests between pairs of groups. Allows us to compare one group with another group. Plain comparisons focus on a few comparisons. |
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Linear contrasts |
Allow us to compare one group or set of groups with another group or set of groups. Can be used with post hoc and a priori tests (though most commonly with a priori). When the weighted sum of treatment means = 0. |
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Linear combination |
A weighted sum of treatment means. Compare average of controls versus average of experimentals. |
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Standard set of coefficients and when you need them |
The absolute values of your coefficient sum to 2. Need when you want to a) interpret a confidence interval and b) calculate effect size. |
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Orthogonal contrasts |
When members of a set of contrasts are independent of one another. The sums of squares of a complete orthogonal contrast = SStreat. Allow you to take SStreat and break it into k-1 variability. |
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Fisher's LSD (least significant difference) requirements |
-Significant F -3 means |
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Bonferroni's inequality |
The probability of occurrence of one or more events can never exceed the sum of their individual probabilities. |
