Assumptions (Fields Ch. 5)

Question 1

Bootstrap

Answer 1

a technique from which the sampling distribution of a statistic is estimated by taking repeated samples (with replacement) from the data set (in effect, treating the data as a population from which smaller samples are taken). The statistic of interest (e.g., the mean, or b coefficient) is calculated for each sample, from which the sampling distribution of the statistic is estimated. The standard error of the statistic is estimated as the standard deviation of the sampling distribution created from the bootstrap samples. From this, confidence intervals and significance tests can be computed.

Question 2

Heterogeneity of variance

Answer 2

the opposite of homogeneity of variance. This term means that the variance of one variable varies (i.e., is different) across levels of another variable.

Question 3

Heteroscedasticity

Answer 3

the opposite of homoscedasticity. This occurs when the residuals at each level of the predictor variables(s) have unequal variances. Put another way, at each point along any predictor variable, the spread of residuals is different.

Question 4

Homogeneity of variance

Answer 4

the assumption that the variance of one variable is stable (i.e., relatively similar) at all levels of another variable.

Question 5

Homoscedasticity

Answer 5

an assumption in regression analysis that the residuals at each level of the predictor variable(s) have similar variances. Put another way, at each point along any predictor variable, the spread of residuals should be fairly constant.

Question 6

Independence

Answer 6

the assumption that one data point does not influence another. When data come from people, it basically means that the behaviour of one person does not influence the behaviour of another.

Question 7

Kolmogorov-Smirnov test

Answer 7

one way to test is the data normal? a test of whether a distribution of scores is significantly different from a normal distribution. A significant value indicates a deviation from normality, but this test is notoriously affected by large samples in which small deviations from normality yield significant results.

Question 8

Levene’s test

Answer 8

this tests the hypothesis that the variances in different groups are equal (i.e., the difference between the variances is zero). It basically does a one-way ANOVA on the deviations (i.e., the absolute value of the difference between each score and the mean of its group). A significant result indicates that the variances are significantly different - therefore, the assumption of homogeneity of variances has been violated. When samples sizes are large, small differences in group variances can produce a significant Levene’s test.

Question 9

Mixed normal distribution

Answer 9

a normal-looking distribution that is contaminated by a small proportion of scores from a different distribution. These distributions are not normal and have too many scores in the tails (i.e., at the extremes). The effect of these heavy tails is to inflate the estimate of the population variance. This, in turn, makes significance tests lack power.

Question 10

Outlier

Answer 10

an observation or observations very different from most others. Outliers bias statistics (e.g., the mean) and their standard errors and confidence intervals.

Question 11

P-P plot

Answer 11

Short for a probability-probability plot. A graph plotting the cumulative probability of a variable against the cumulative probability of a particular distribution (often a normal distribution). Like a Q-Q plot, if values fall on the diagonal of the plot then the variable shares the same distribution as the one specified. Deviations from the diagonal show deviations from the distribution of interest.

Question 12

Parametric test

Answer 12

a test that requires data from one of the large catalogue of distributions that statisticians have described. Normally this term is used for parametric tests based on the normal distribution, which require four basic assumptions that must be met for the test to be accurate: a normally distributed sampling distribution (see normal distribution), homogeneity of variance, interval or ratio data, and independence.

Question 13

Q-Q plot

Answer 13

short for a quantile-quantile plot. A graph plotting the quantiles of a variable against the quantiles of a particular distribution (often a normal distribution). Like a P-P plot, if values fall on the diagonal of the plot then the variable shares the same distribution as the one specified. Deviations from the diagonal show deviations from the distribution of interest.

Question 14

Robust test

Answer 14

a term applied to a family of procedures to estimate statistics that are reliable even when the normal assumptions of the statistic are not met.

Question 15

Transformation

Answer 15

the process of applying a mathematical function to all observations in a data set, usually to correct some distributional abnormality such as skew or kurtosis.

Question 16

Trimmed mean

Answer 16

a statistic used in many robust tests. It is a mean calculated after a certain percentage of the distribution has been removed at the extremes. For example, a 20% trimmed mean is a mean calculated after the top and bottom 20% of ordered scores have been removed. Imagine we had 20 scores representing the annual income of students (in thousands, rounded to the nearest thousand: 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 35. The mean income is 5 (£5000). This value is biased by an outlier. A 10% trimmed mean will remove 10% of scores from the top and bottom of ordered scores before the mean is calculated. With 20 scores, removing 10% of scores involves removing the top and bottom 2 scores. This gives us: 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, the mean of which is 3.44. The mean depends on a symmetrical distribution to be accurate, but a trimmed mean produces accurate results even when the distribution is not symmetrical. There are more complex examples of robust methods such as the bootstrap.

Question 17

Weighted least squares

Answer 17

a method of regression in which the parameters of the model are estimated using the method of least squares but observations are weighted by some other variable. Often they are weighted by the inverse of their variance to combat heteroscedasticity.

Question 18

What are the tests that I can use to see if my data is normal?

Answer 18

K-S test, S-W test, Chi-Squared test. Can use the free software.

Question 19

What are the assumptions of parametric data?

Answer 19

Normally distributed data (either the sampling distribution or the errors in the model must be normal)

Homogeneity if variance

Interval data

Independence

Question 20

As the sample gets larger, we can be more confident that…

Answer 20

The sampling distribution will be normally distributed (samples of 30 or more) - important for first assumption of parametric data tests (all the tests we do)