Economics 5550/6550

Statistical Tools

2014

Statistics and Hypotheses

• If you are following the health care system, you see a lot of discussion. Let's consider health care costs. Most simply:

• How fast are they rising?• Are ours higher than Canada's, UK, or others; if so how

much?• Does managed care reduce costs; if so, how much?• Are patients in managed care settings sicker, healthier;

if so how much?

Most of the time, such assertions are made with certainty. Yet, they are almost certainly not certain. These involve testing hypotheses.

Many of you have had some passing experience with experimental laboratory work. Most often the work is a demonstration, rather than an experiment (i.e. mix up these 2 chemicals and you'll blow up the lab).

Let's see what we have in mind with empirical work.

Statistics and Hypotheses

Toggle to Country Data

• Pct GDP Spent Over Time (EXCEL SS)

• Look at the Data

Percent of GDP Spent on Health Care, 1960-2010

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

Year

Per

cen

t

Canada

France

Germany

Japan

United Kingdom

United States

What does this look like to you?

How do we formulate hypotheses?

The style is to use the "null" hypothesis; then contrast it to the "alternative" hypothesis. Null means “nothing is happening.”

Let's look at several countries. Are the health expenditure shares equal?

H0 : sUS = sUK.

H1 : sUS sUK.

H0 : sUS = sUK.

H1 : sUS sUK.

So, if sUS seems to be about the same as sUK, we'll accept the null hypothesis. If it seems to be different, we'll reject the null hypothesis. This type of hypothesis is called a "simple hypothesis."

Simple Hypotheses

Composite Hypotheses

Often we want to consider more than simply equality. Suppose we are examining the health expenditures of rich people, Er, as opposed to poor people, Ep. We could certainly compare equalities. We might also wish to test the received wisdom that the rich spend more, or:

H0: Er = Ep

H1: Er > Ep

Composite Hypotheses

H0: Er = Ep

H1: Er > Ep

As opposed to a "simple hypothesis" above, this called a "composite hypothesis." Note that this is not the same as above. If somehow, we discovered that Ep > Er, it would not validate this hypothesis.

So what? Well, let's go back to the health care system, for example. What costs more? What is more effective? Ultimately, statistical analysis is how the linkages between smoking and cancer were discovered. Tobacco industry fought this for years!

So, how do we test hypotheses?

• Let’s look at the US v. the UK• Is the share different?• It was in 1960.• That’s only one data point. Let’s look at

some others.• It looks like most of the time US > UK.

Measure of central tendency• There is an enormous body of statistical literature

that suggests that if we take the average, or arithmetic mean, of a group of people, we will have a better measure of "central tendency." Think GPA.

• The intuition is straightforward. While any single observation may be suspect, the premise is that a measure taken from a larger number of observations may be less suspect.

• What other measures are sometimes used?• Median, mode. Why?

Hypothesis testing

• Here (Spreadsheet):

10.6; 6.0US UKs s

Hypothesis testing

• Well, they look different. Hard to imagine that they would be identical. Are they different?

• d = 10.6 – 6.0 = 4.6. Is that little or big, compared to 0?

10.6; 6.0US UKs s

Testing Differences

• Let's see if we can do things a bit more rigorously, looking at the "spread" of the data.

• So:

US UKd s s

Testing Differences

• We'd like to determine whether the difference d could have been generated randomly; that is, what is the possibility that in truth sUS = sUK, and somehow we just managed to find 2 samples that generated these separate means?

• We want to compare the difference to the spread. If the difference is large (relative to the spread), it would seem to be unlikely that it was caused at random. If it is small, it might have been caused at random.

US UKd s s

Testing Differences

• A useful way, and by far the most frequent way, that we calculate the "spread" is to subtract each value from the mean, square the term, add the squared terms, and divide by the number of observations.

2 2 21 2( ) ( ) ...( )nx x x x x x

VN

Testing Differences

• If the spread is large, then this measure is large. If the spread is small, then this measure will be small. We often take the square root of this difference to get the standard error for the US. We can do the same for the UK, as well.

• Our EXCEL worksheet helps us out.

Testing Differences

• It turns out that we can calculate a standard error of the difference which equals:

• sdd = (sdUS2 + sdUK

2)1/2

• sdd = (.20382 + .52072)1/2 = 0.5592.

• Let's compare the spread, or standard error, of the difference to the difference. The difference of 4.6 is about 8.3 times as big as the standard error. Clearly, if the difference was 0, we'd accept the fact that sUS = sUK.

t statistic

The ratio of the difference to the standard error is called a Student’s t statistic.

You used to have to look these up with tables, but now we can actually use a spreadsheet.

Let’s take a look.

What I learned on this summer on vacation

• “Student” was William Sealy Gossett, a statistician for Guiness.

• Gosset applied his statistical knowledge – both in the brewery and on the farm – to the selection of the best yielding varieties of barley. 

• Another researcher at Guinness had previously published a paper containing trade secrets of the Guinness brewery.

• To prevent further disclosure of confidential information, Guinness prohibited its employees from publishing any papers regardless of the contained information.

• However, after pleading with the brewery and explaining that his mathematical and philosophical conclusions were of no possible practical use to competing brewers, he was allowed to publish them, but under a pseudonym ("Student"), to avoid difficulties with the rest of the staff. Thus his most noteworthy achievement is now called Student's, rather than Gosset's, t-distribution.

Key points for hypothesis testing:

State hypotheses clearly

Choose suitable sample

Calculate appropriate measures of central tendency and dispersion

Draw appropriate inferences