10.3: Use and Abuse of Tests Carrying out a z-confidence interval calculation is simple. On a TI-83, it’s STAT-->TESTS-->7 Similarly, carrying out a z-signficance test is simple. On a TI-83, it’s STAT-->TESTS-->1 The trick is knowing when it’s APPROPRIATE to use these tools. Don’t get calculator happy! Thinking is required.

Choosing a Level of Significance Recall that the P -value gives you the strength of the evidence against the null hypothesis. We often decide whether or not to reject the null hypothesis based on the chosen significance level, a. However, a should not be treated as some sort of universal standard. Different situations require difference significance levels.

Choosing a Level of Significance The significance levels of 1%, 5%, 10% come from a time before the help of computers or software. They were born from convenience, not scientific rigor. Because of this, no matter what the signifiance level is, YOU SHOULD ALWAYS REPORT THE P -value!!!

Choosing a Level of Significance Another thing to consider: There is NEVER an exact cut-off between significant and insignificant, only increasing evidence against the null hypothesis as the P -value decreases. For example, there is no practical difference between P -values of and 0.051, even though the first is significant at the 5% level and the second, technically, is not. You have to use your discretion!

What statistical significance doesn’t mean Suppose we perform a significance test and determine significance. (P-value ≤ a ). This means: there is good evidence that an effect is present. This DOES NOT mean: there is good evidence that the effect is large.

What statistical significance doesn’t mean Given a large enough sample, even small deviations from the mean will yield significance. For example, if n = 1000, a correlation of r = 0.08 is significant at the 1% level given a null hypothesis of r = 0. But r = 0.08 is an extremely small correlation! So what we’ve shown is: There DOES appear to be a positive association. What we HAVEN’T shown is: The association is large.

What statistical significance doesn’t mean Q: So what do we do? A: PLOT YOUR DATA! Look for outliers, influentials, or any systematic deviations (bias). A significance test is a cool thing, but it’s overused. While a significance level says NOTHING about the size of an effect, a Confidence Interval DOES help you determine the size of an effect. IT NEVER HURTS TO USE BOTH!

Proper Steps of Data Analysis Use graphical analysis - NQPs, Boxplots, and Histograms. Use significance tests to determine if there is an effect. Use confidence intervals to determine the size of the effect.

Don’t ignore lack of significance It is easy to get obsessed with trying to show that an effect exists. Upon discovering there isn’t one, many researchers have discarded their work. But keep in mind: not finding an effect where we expected to see one is important too! Moral: ALWAYS REPORT YOUR RESULTS For one, you don’t want to doom others to doing your experiment. For another, it is worthwhile to analyze WHY no effect was detected.

Statistical Inference is not valid for all data sets Remember, badly designed experiments produce worthless results. Even math can’t save the day. Remember also that we sometimes don’t always know WHAT caused a result, even in a seemingly well-designed experiment. This raises the issues of confounding and lurking variables.

Statistical Inference is not valid for all data sets For example, there exists statistical significance between the English vocabulary scores of high school seniors who have studied a foreign language and those who have not. Does this mean taking a foreign language cause higher English vocabulary scores? Or does it mean that taking a foreign language is confounded by the type of student who takes a foreign language during their senior year? Statistical significance tells us a difference exists, but it doesn’t tell us HOW or WHY the difference exists.

Beware of searching for significance Statistical Significance has its basis in probability. But even randomness can show a pattern if you look long enough. This can lead to problems if you perform too many tests on the same data.

Beware of searching for significance Example: There was a team of psychiatrists who performed SEVENTY-SEVEN significance tests on a set of data that contained 77 variables! Consider this, if you made 77 tests at the 5% level, you would expect a few to be significant by pure chance. After all, results significant at the 5% level happen 5 out of 100 times (in the long run) EVEN WHEN THE NULL HYPOTHESIS IS TRUE! So it wasn’t really meaningful when the psychiatrists achieved significance for two of the tests.

Morals of the Story A significance level should be used as a guideline, not a rigid barrier. Significance levels should vary based on the severity of the experiment.

Morals of the Story Significance tests DO NOT tell you whether an effect is large or small. They only tell you whether an effect exists. Use Confidence Intervals to determine the size of an effect.

Morals of the Story Don’t ignore lack of significance. Sometimes you learn more from failing than you do from succeeding! Beware of confounding and lurking variables before drawing any conclusions from a significance test. Searching for significance should NOT be the goal. It is simply a tool which can yield helpful information for a properly designed experiment.