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Kindle Notes & Highlights
by
Andy Field
Read between
January 9 - February 23, 2021
If you have a lot of scores Q-Q plots can be easier to interpret than P-P plots because they display fewer values.
If you must use the K-S test, its statistic is denoted by D and you should report the degrees of freedom (df) in brackets after the D.
When predictor variables are formed of categories, if you decide that you need to check the assumption of normality then you need to do it within each group separately
The reason for looking at the assumption of linearity and homoscedasticity together is that we can check both with a single graph. Both assumptions relate to the errors (a.k.a. residuals) in the model and we can plot the values of these residuals against the corresponding values of the outcome predicted by our model in a scatterplot. The resulting plot shows whether there is a systematic relationship between what comes out of the model (the predicted values) and the errors in the model.
When these assumptions have been violated you won’t see these exact patterns, but hopefully these plots will help you to understand the general anomalies to look out for.
homoscedasticity/homogeneity of variance means that as you go through levels of one variable, the variance of the other should not change.
SPSS produces something called Levene’s test (Levene, 1960), which tests the null hypothesis that the variances in different groups are equal.
Some people also look at Hartley’s Fmax also known as the variance ratio (Pearson & Hartley, 1954). This is the ratio of the variances between the group with the biggest variance and the group with the smallest. This ratio was compared to critical values in a table published by Hartley.
Levene’s test, which can be based on differences between scores and the mean or between scores and the median. The median is slightly preferable (because it is less biased by outliers).
The variances are practically equal. So, why does Levene’s test tell us they are significantly different? The answer is because the sample sizes are so large: we had 315 males and 495 females, so even this very small difference in variances is shown up as significant by Levene’s test
Levene’s test can be reported in this general form: F(df1, df2) = test statistic, p = p-value.
Trim the data: Delete a certain quantity of scores from the extremes. Winsorizing: Substitute outliers with the highest value that isn’t an outlier. Apply a robust estimation method: A common approach is to use bootstrapping. Transform the data: Apply a mathematical function to scores to correct problems.
Probably the best of these choices is to use robust tests, which is a term applied to a family of procedures to estimate statistics that are unbiased even when the normal assumptions of the statistic are not met
Trimming the data means deleting some scores from the extremes.
trimming involves removing extreme scores using one of two rules: (1) a percentage based rule; and (2) a standard deviation based rule.
If you take trimming to its extreme then you get the median, which is the value left when you have trimmed all but the middle score.
we calculate the mean in a sample that has been trimmed in this way, it is called (unsurprisingly) a trimmed mean.
rather than the researcher deciding before the analysis how much of the data to trim, an M-estimator determines the optimal amount of trimming necessary to give a robust estimate of, say, the mean.
Standard deviation based trimming involves calculating the mean and standard deviation of a set of scores, and then removing values that are a certain number of standard deviations greater than the mean.
Winsorizing the data involves replacing outliers with the next highest score that is not an outlier.
best option if you have irksome data (other than sticking a big samurai sword through your head) is to estimate parameters and their standard errors with methods that are robust to violations of assumptions and outliers.
use methods that are relatively unaffected by irksome data.
The first we have already looked at: parameter estimates based on trimmed data such as the trimmed mean and M-estimators. The second is the bootstrap
Bootstrapping gets around this problem by estimating the properties of the sampling distribution from the sample data.
The idea behind transformations is that you do something to every score to correct for distributional problems, outliers, lack of linearity or unequal variances.
If you do decide to transform scores, use the compute command, which enables you to create new variables.
Create new variables from existing variables:
Create new variables from functions:
Our starting point with a correlation analysis is, therefore, to look at scatterplots of the variables we have measured.
Remember that the variance of a single variable represents the average amount that the data vary from the mean. Numerically, it is described by: (8.4)
when one variable deviates from its mean we would expect the other variable to deviate from its mean in a similar way.
When we multiply the deviations of one variable by the corresponding deviations of a second variable, we get the cross-product deviations.
positive covariance indicates that as one variable deviates from the mean, the other variable deviates in the same direction.
a negative covariance indicates that as one variable deviates from the mean (e.g., increases), the other deviates from the mean in the opposite direction (e.g., decreases).
the covariance depends upon the scales of measurement used: it is not a...
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To overcome the problem of dependence on the measurement scale, we need to convert the covariance into a standard set of units. This process is known as standardization.
standardized covariance is known as a correlation coefficient and is defined as follows: (8.7) in which sx is the standard deviation of the first variable and sy is the standard deviation of the second variable (all other letters are the same as in the equation defining covariance). This coefficient, the Pearson product-moment correlation coefficient or Pearson’s correlation coefficient, r, was invented by Karl Pearson with Florence Nightingale David2
We have just described a bivariate correlation, which is a correlation between two variables.
The first is to use the trusty z-scores that keep cropping up in this book.
z-scores are useful because we know the probability of a given value of z occurring, if the distribution from which it comes is normal.
one problem with Pearson’s r, which is that it is known to have a sampling distribution that ...
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we can adjust r so that its sampling distribution is normal: (8.8) The resulting zr has a sta...
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the hypothesis that the correlation coefficient is different from 0 is usually (SPSS, for example, does this) tested not using a z-score, but using a different test statistic called a t-statistic with N − 2 degrees of freedom. This statistic can be obtained directly from r: (8.11)
a 95% confidence interval is calculated (see Eq. 2.15) as: (8.12)
our transformed correlation coefficients, these equations become: (8.13)
we can convert back to a correlation coefficient using: (8.14)
Ranking the data reduces the impact of outliers. Furthermore, given that normality matters only for inferring significance and computing confidence intervals, we could use a bootstrap to compute the confidence interval, then we don’t need to worry about the distribution.
because the confidence intervals are derived empirically using a random sampling procedure (i.e., bootstrapping) the results will be slightly different each time you run the analysis.
