# Significance Tests

A *significance test* is a way of working out if a particular *difference* is likely to be meaningful or be a fluke.

## Contents

## Sampling error

Imagine doing a study of 200 consumers and finding that 41% said that Coca-Cola was their favorite soft drink. Now, imagine you did another study and found that in the next study 40% of people preferred Coca-Cola. And, imagine you did a third study and found that 43% of people preferred Coca-Cola. What can you infer from the differences between these studies? There are three explanations:

*Explanation 1:* The world changed in some way between each of these studies and the proportion of people preferring Coca-Cola dropped a little and then increased (i.e., moved from 41% to 40% and then up to 43%).

*Explanation 2:* The difference between the two studies is just random noise. More specifically, as each study only sampled 200 people it is to be expected that we should get small differences between the results of these. Or, to use the jargon, there is *sampling error.*^{[note 1]}

**Explanation 3: **A mix of explanations 1 and 2.

## Significance tests

A *significance test* is a rule of thumb that is used to help to determine whether a difference between two numbers is likely to reflect a meaningful difference in the world at large (i.e., explanation 1 above), or, is merely a fluke caused by sampling error (i.e., explanation 2).

There are many thousands of different significance tests with exotic names like Wilk's lambda, Fisher's Exact Test and so on. However, when analyzing survey data there is generally no need to go into such specific detail about which test to use and when as most significance tests that are applied when analyzing real-world surveys are either *exception* tests or *column comparisons*.

### Exceptions tests

Consider the following chart from Displayr. Reading across the `Coca-Cola` row we can see that:

- 65% of people aged 18 to 24 prefer Coca-Cola.
- 41% of people aged 25 to 29 prefer Coca-Cola.
- 43% of people aged 30 to 49 prefer Coca-Cola.
- 40% of people aged 50 or more prefer Coca-Cola.

That we get different results in each of the age groups is to be expected. The process of selecting people to participate in a survey means that by chance alone we expect that we will get slightly different results in the different age groups even if it was the case that there really is no difference between the age groups in terms of preference for Coca-Cola (i.e,. due to *sampling error*). However, the level of preference for 18 to 24 year olds is substantially higher. In the chart below the font color and the arrow indicate that this result is *significantly high*. That is, because the result has been marked as being *statistically significant*, the implication is that the much higher result observed for the 18 to 24s is not merely a fluke and signifies that in the population at large it is true that 18 to 24 year olds have a higher level of preference for Coca-Cola.

Looking elsewhere on the table we can see that: Diet Coke preference seems to be low among people aged 18 to 24, Pepsi scores relatively well among the 30 to 49 year olds and so on. Each of these results are examples of *exception tests*, which are statistical tests that identify results that are, in some way, exceptions to the norm.^{[note 2]}

Now look carefully at the row for `Coke Zero`. The score for the 18 to 24 year olds is less than half that of the other age groups. However, it is not marked as being significant and thus the conclusion is that the relatively low score for 18 to 24 year olds *may* be a fluke and does not reflect a true difference in the population at large. The word 'may' has been italicized to emphasize a key point: there is no way of known for sure whether the low score among the younger people in the survey reflects a difference in the population at large or is just a weird result that occurs in this particular sample. Thus, all significance tests are just guides they rarely prove anything and we always need to apply some commonsense when interpreting them.

### Column comparisons

The table below shows exactly the same data from the same survey as shown above. However, whereas the chart above showed results as *exceptions*, this one instead shows a more complicated type of significance test called *column comparisons*. Each of the columns is represented by a letter, shown at the bottom of the page. Some of the cells of the table contain letters and these indicate that the result in the cell is significantly higher the results in the columns that are listed. For example, looking at the `Coca-Cola` row, the appearance of `b c d` indicates that the preference for Coca-Cola of 65% among the 18 to 24 year olds is significantly higher than the preference scores of the 25 to 29 year olds (`b`), the 30 to 49 year olds (`c`) and the people aged 50 or more (`d`). That the letter are in lowercase tells us that the difference is not super-strong (in the exception shown above, the length of the arrows communicates the degree of statistical significance).

Note that while many of the conclusions that we can get from this table are similar to those from the chart above, there are some differences. For example, in the chart above we drew the conclusion that the Diet Coke preference was significantly lower among the 18 to 24 year olds than among the population at large. However, the column comparisons tell us only that the 18 to 24s have a lower score than the 30 to 49s (i.e., we know this because the `a` for the 30 to 49s tells us that they have a stronger preference than the 18 to 24s who are represented as column `a`).

Why do the two ways of doing the tests get different results? There are some technical explanations.^{[note 3]} But all the really amount to is this: the different approaches make slightly different technical results and, consequently, they get slightly different results. An analogy that is useful is to think about different ways of reporting news: we can get the same story reported on TV, in a newspaper and in a blog and each way will end up emphasising slightly different aspects of the truth.

## The determinants of statistical significance

There are many different factors that influence whether a particular difference is reported as being statistically significant or not, including:

- The size of differences being compared (i.e., the bigger the difference the more likely it will be significant). This is exactly the same idea that is discussed on in the page on Determining The Sample Size.
- The sample size. Differences observed in larger sample sizes are more likely to be statistically significant.
- The specific
*confidence level*of the testing. - The number of technical assumptions that are made in the test (e.g., assumptions of
*normality*). In the main, the fewer assumptions that are made the lower the chance that a result is concluded as being statistically significant. - If and how the data has been weighted. The greater the effect of the weighting the less likely that results will be statistically signicant.
- The number of tables that are viewed and the size of the tables. The greater the amount of analyses that are viewed, the greater the chance of fluky results.
- How the data has been collected (in particular, what approach to
*sampling*was adopted). - The technical proficiency of the person that has written the software conducting the test. In particular, most formulas presented in introductory statistical courses only take into account the first three of the issues listed above and most commercial programs deal with the weighting incorrectly. The general ambiguity of statistical testing in terms of it not being able to give definitive conclusions combined with the large number of technical errors that are made in practical applications of significant tests again lead to the same conclusion presented earlier: statistical tests are nothing more and nothing less than a useful way of identifying interesting results that may reflect how the world works but also may just be weird flukes.

## A more technical explanation

The above description is not a simplification. It is an accurate depiction of what statistical testing does and is. A more theoretical discussion of how to interpret statistical tests on tables is at Statistical Tests on Tables at SurveyAnalysis.org.

## Notes

- ↑ Or, to be more precise, sampling error is the difference between what we observe in a random sample and what we would have obtained had we interviewed in the population.
- ↑ The term
*exception test*is not a standard term. The closest there is to a standard term for such a test is*studentized residuals in contingency tables*, but even this is a pretty obscure term. - ↑ In particular, the
*exceptions*test has more statistical power due to the pooling of the sample, the columns comparisons are not transitive and there are smaller sample sizes for column comparisons than for exception tests.