Experimental Design Basics

Variables

Variables in statistics are a lot like variables in algebra or calculus, except that they are typically thought of as having a distribution of values. For instance, a variable might describe someone's reaction time to a stimulus, but this variable it will be different each time it is measured. So a variable like reaction time can be understood as a distribution---a collection of all possible reaction times and how likely each is---rather than a single value.

Continuous vs. Discrete

Variables in experiments can be continuous or discrete. In psychology experiments, continuous variables are almost always real numbers (e.g. people's height, reaction time). Discrete variables are ones that do not vary continuously. Typical examples in psychology experiments take on only a finite number of values, for instance in measures like level of education, number of children, or correct/incorrect. However, it is possible that discrete variables give a potentially infinite number of values, as in counts of how neuron spikes occur in a given amount of time.

Dependent vs. Independent

The terms dependent variable and independent variable are used to distinguish between two types of quantifiable factors being considered in an experiment. In simple terms, the independent variable is typically the variable representing the value being manipulated or changed and the dependent variable is the observed result of the independent variable being manipulated.

In simple experiments, you manipulate one variable and measure another (hopefully while holding everything else constant). For instance, you might measure people's performance on a test (i) when classical music is playing versus (ii) when no music is playing. Here, the independent variable (the thing you manipulate) is whether or not music is play (i/ii) and the dependent variable (the thing you care about) is performance on the test.

Within vs. Across Subjects

It's important to realize that independent variables can be either within subjects or across subjects. When an independent variable is within subjects, it means that you have measured each subject with each level of the variable---for instance, if you each participant took the test with and without music. An across subjects design would give each participant only one test, either with or without music.

In general, within subject designs are more likely to find effects because they control for additional noise---in this case, each subject's typical ability to answer questions on the test. Here is another example of why within designs are more powerful: suppose you were trying to determine if 4th graders were taller than 3rd graders. If you took a sample of a typical 4th grade and a typical 3rd grade class, you would see a highly overlapping distribution of heights. That is, each classes' heights would be very variable and there would not be much difference in the mean heights, so it would take a lot of samples to find a difference. But suppose you performed the within subjects experiment. You could take 3rd graders and measure their heights, and then waited a year until they were 4th graders, and measured their heights again. Everyone would have grown and you could very easily find a significant result by comparing each child's heights in 3rd and 4th grade. This is a within subjects design because each subject is measured twice, once in each condition (3rd vs. 4th grade), and is clearly much more likely to find an effect that is real.

Samples and Summary Statistics

Typically in experiments, one will measure a variable some number of times to collect a sample. For instance, if we were interested in the heights of human adults, we might measure the heights of a million people from New York. This would give us a sample of different human adult heights. Sets of samples are hard to understand on their own---what can a million individual values tell us on their own?

Summary statistics provide a way to capture the basic trends in a sample of data in a concise, informative way. Probably the most common summary statistic is a mean: the mean of a set of numbers gives the average (intuitively, the typical value) of the sample. Another common measure is the variance, which computes the variability in the sample. So the mean of the heights of everyone in New York state would capture the typical value of people's height and the variance of heights would tell you how much variability there is between people.

[To come: a word on populations.]

Estimators

We can view measures like the mean and variance of a sample as estimators of some true, unknown property of the population you care about. So the mean computed on the sample estimates (or approximates) the true mean of the population. Since we usually want to make statements about the true state of the world (men are taller than women) rather than our sample (our sample of men is taller than our sample of women), its useful to think about using our sample to estimate or approximately measure some true property of the world.

Estimates can be biased or unbiased. Biased estimators are ones which, intuitively, are expected to give a (perhaps slightly) wrong answer. Unbiased estimators are expected to give the correct answer. As an example, suppose you collected a sample of heights and for some reason threw out the shortest 10 people before computing the mean. The mean you compute will be a biased estimator of the true mean since it will tend to overestimate people's typical height. But, as you get more and more people, the shortest 10 will matter less and less and so the amount of bias will decrease as you get more and more people.

For computing variance (or standard deviation), you should remember to use the unbiased estimator of variance, which includes a N-1 instead of an N in the denominator.

Choose the Right Statistical Test

In statistical inferences, the "true" populations that you care about are usually assumed to be distributed in a particular way. The most common assumption is that data items in the populations are distributed according to a normal distribution (i.e. the bell curve, also referred to as Gaussian distributions). Nearly all statistical tests you have been taught in an undergrad statistics class require this assumption to hold. When this assumption does not hold (for example, when your data are not interval variables, you need to think twice about which test to use), another set of tests, which are referred to as non-parametric tests, should be used. Although in most BCS lab courses, encountering a situation where the normality assumption does not hold is rare, keep in mind that the t-test, for instance, does not apply everywhere. The following table may be helpful in ensuring you choose the correct statistical test for your experimental data.

Interpret a Significant Result

Statistical significance means something very specific in experimental work: an effect is significant if the test statistic you find is very unlikely to have occurred under the null hypothesis. For instance, if you run a t-test and find a t-value of 5.8, this is extremely unlikely to occur when the null hypothesis (no difference in means) is true. The interpretation of this is that the null hypothesis is unlikely to be correct. The p-value measures what proportion of the time the null hypothesis will generate a test statistic at least as large as the one you see. So if the p-value is 0.05, it means that 5% of the time---1 in 20 times---the null hypothesis will generate a test statistic at least as large as the one you see. So in that sense, the p-value provides an intuitive measure for how unlikely the null hypothesis is to have generated data like the data you observe. But be careful--it is possible that the null hypothesis is actually true; it is just statistically unlikely.

The term statistically significant does not mean that the result is significant in the sense of being important. A result can be statistically significant (the test statistic is unlikely under the null) but really not be that important. For instance, people's attractiveness might have a statistically significant effect on income, but this effect might not be that important if income is primarily determined by other factors like job type and education level.

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