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Modern Approaches

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Modern Approaches

The Bootstrap with Inferential Example

Non-parametric or distribution-free tests have more lax and/or different assumptions

Properties:

No assumption about the underlying distribution being normal

More sensitive to medians than means (which is good if you’re interested in the median)

Some may not be very affected by outliers

Rank tests

Parametric tests will typically be used when assumptions are met as they will usually have more power

Though the non-parametric tests might be able to match that power under certain conditions

Non-parametric tests are often used with small samples or violations of assumptions

- For purely categorical situations
- Chi-square and Loglinear analysis

- Rank-based approaches to help with normality and outliers
- Wilcoxon t for independent and dependent samples, Mann-Whitney U
- Kruskal-Wallis, Friedman for more than 2 groups

- Transformations of data
- Logarithmic, reciprocal etc.

The Bootstrap

The basic idea involves sampling with replacement from the sample data to produce random samples of size n

Each of these samples provides an estimate of the parameter of interest

Repeating the sampling a large number of times provides information on the variability of the estimate i.e. its standard error

Necessary for any inferential test

How many hours of TV watched yesterday

1000 samples

Distribution of Means of each sample

Mean = 3.951

Inferential Use

Bootstrapping applied to t-tests

- Wilcox notes that when we sample from a non-normal population, assuming normality of the sampling distribution maybe optimistic without large samples
- Furthermore, outliers have an influence on both the mean and sd used to calculate t
- Actually has a larger effect on variance, increasing type II error due to std error increasing more so than the mean

- This is not to say we throw the t-distribution out the window
- If we meet our assumptions and have ‘pretty’ data, it is appropriate

- However, if we cannot meet the normality assumption we may have to try a different approach
- E.g. bootstrapping

- In the two-sample case we have an additional assumption (along with normality and independent observations)
- We assume that there are equal variances in the groups
- Recall our homoscedasticity discussion

- Often this assumption is untenable, and the results, like other violations result in using calculated probabilities that are inaccurate
- Can use a correction, e.g. Welch’s t

- It is one thing to say that they are unequal, but what might that mean?
- Consider a control and treatment group, treatment group variance is significantly greater
- While we can do a correction, the unequal variances may suggest that those in the treatment group vary widely in how they respond to the treatment
- Another reason for heterogeneity of variance may be related to an unreliable measure being used*
- No version of the t-test takes either into consideration
- Other techniques, assuming enough information has been gathered, may be more appropriate (e.g. hierarchical), and more reliable measures may be attainable

- The good
- If assumptions are met, t-test is fine
- When assumptions aren’t met, t-test may still be robust with regard to type I error in some situations
- With equal n and normal populations HoV violations won’t increase type I much
- With non-normal distributions with equal variances, type I error rate is maintained also

- The bad
- Even small departures from the assumptions result in power taking a noticeable hit (type II error is not maintained)
- t-statistic, CIs will be biased

- Recall the notion of a sampling distribution
- We never have the population available in practice, so we take a sample (one of an infinite amount of possible ones)
- The sampling distribution is a theoretical distribution whose shape we assume

- Here we will do as before where the basic idea involves sampling with replacement from the sample data (essentially treating it as the population) to produce random samples of size n
- We create an empirical sampling distribution rather than assuming a theoretical one

- Each of these samples provides an estimate of the parameter of interest
- Repeating the sampling a large number of times provides information on the variability of the estimator

- Hypothetical situation:
- If we cannot assume normality, how would we go about getting a confidence interval?
- Wilcox suggests that assuming normality via the central limit theorem doesn’t hold for small samples, and sometimes could require as much as 200* to maintain type I error if the population is not normally distributed
- If we do not maintain type I error, confidence intervals and inferences based on them will be suspect

- How might you get a confidence interval for something besides a mean?

- If we cannot assume normality, how would we go about getting a confidence interval?
- Solution:
- Resample (with replacement) from our own data based on its distribution
- Treat our sample as a population distribution and take random samples from it

- We will start by considering a mean
- We can bootstrap many sample means based on the original data
- One method would be to simply create this distribution of means, and note the percentiles associated with certain values

- Here are some values (from Wilcox text), mental health ratings of college students
- Mean = 18.6
- Bootstrap mean (k =1000) = 18.52

- The bootstrapped 95% CI is
- 13.85, 23.10

- Assuming normality
- 13.39, 23.81

- Different coverage (non-symmetric for bootstrap), and the classical approach is noticeably wider

2,4,6,6,7,11,13,13,14,15,19,23,24,27,28,28,28,30,31,43

- Another approach would be to create an empirical t distribution
- Recall the formula for a one-sample t
- For our purposes here, we will calculate a t, 1000 times, as follows. With each mean and standard deviation of 1 of those 1000 samples, calculate

- This would give us a t distribution with 1000 t scores
- What we would now do for a confidence interval is find the exact t corresponding to the appropriate quantiles (e.g. .025,.975), and use those to calculate a CI using the original sample statistics

- So what we have done is, instead of assuming some sampling distribution of a particular shape and size, we’ve created it ourselves and derived our interval estimate from it
- Simulations have shown that this approach is preferable for maintaining type I error with larger samples in which the normality assumption may be untenable.

- Comparing independent groups
- Step 1 compute the bootstrap mean and bootstrap sd as before, but for each group
- Each time you do so, calculate T*
- This again creates your own t distribution.

- Use the quantile points corresponding to your confidence level in computing your confidence interval on the difference between means, rather than the tcv from typical distributions
- Note however that your T* will not be the same for the upper and lower bounds
- Unless your bootstrap distribution was perfectly symmetrical
- Not likely to happen, so…

- One can obtain ‘symmetric’ intervals
- Instead of using the value obtained in the numerator (mean-mu) or (diff b/t means – mu1-mu2), use its absolute value
- Then apply the standard + formula
- This may in fact be the best approach for most situations

- We can incorporate robust measures of location rather than means
- Eg. Trimmed means

- With a program like R it is very easy to do both bootstrapping and with robust measures using Wilcox’s libraries
- http://www-rcf.usc.edu/~rwilcox/
- Put the Rallfun files (most recent) in your version 2.x main folder and ‘source’ them, then you’re ready to start using such functionality
- E.g. source(“C:/R/Rallfunv1.v5”)
- Example code on last slide

- The general approach can also be extended to more than 2 groups, correlation, and regression

- Accuracy and control of type I error rate
- As opposed to just assuming that it’ll be ok

- Most of the problems associated with both accuracy and maintenance of type I error rate are reduced using bootstrap methods compared to Student’s t
- Wilcox goes further to suggest that there may be in fact very few situations, if any, in which the traditional approach offers any advantage over the bootstrap approach
- The problem of outliers and the basic statistical properties of means and variances as remain however

- source("Rallfunv1-v7")
- source("Rallfunv2-v7")
- y=c(1,1,2,2,3,3,4,4,5,7,9)
- z=c(1,3,2,3,4,4,5,5,7,10,22)
- t.test(y,z, alpha=.05)
- yuenbt(y,z,tr=.20,alpha=.05,nboot=600,side=T)