1 11 00 This sequence explains the logic behind a one-sided t test. ONE-SIDED t TESTS 0 +sd 0...

1

10

This sequence explains the logic behind a one-sided t test.

ONE-SIDED t TESTS

0+sd0–sd0–2sd 1+2sd

2.5% 2.5%

Xf

X

null hypothesis: H0: = 0

alternative hypothesis: H1: = 1

2

We will start by considering the case where can take only two possible values: 0, as under the null hypothesis, and 1, the only possible alternative.

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

2.5% 2.5%



3

An example of this situation is where you are selling computer accessories and among your products there are two types of removable lap-top batteries: regular (average charge duration 0) and long life (average charge duration 1).

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

2.5% 2.5%



4

You have been sent an unmarked shipment and you take a sample and see how long they last. Your null hypothesis is that they are regular batteries.

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

2.5% 2.5%



5

Suppose that the sample outcome is as shown. You would not reject the null hypothesis because the sample estimate lies within the acceptance region for H0.

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

2.5% 2.5%



6

Here you would reject the null hypothesis and conclude that the shipment was of long life batteries.

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

2.5% 2.5%



7

Here you would stay with the null hypothesis.

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

2.5% 2.5%



8

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

2.5% 2.5%

A sample outcome such as this one gives rise to a serious problem. It lies in the rejection region for H0, so our first impulse would be to reject H0.



9

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

2.5% 2.5%

But to reject H0 and go with H1 is nonsensical. Granted, the sample outcome seems to contradict H0, but it contradicts H1 even more strongly.



10

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

2.5% 2.5%

The probability of getting a sample outcome like this one is much smaller under H1 than it is under H0.



11

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

2.5%

For this reason the left tail should be eliminated as a rejection region for H0. We should use only the right tail as a rejection region.



12

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

2.5%

The probability of making a Type I error, if H0 happens to be true, is now 2.5%, so the significance level of the test is now 2.5%.



13

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

5%

We can convert it back to a 5% significance test by building up the right tail until it contains 5% of the probability under the curve. It starts 1.645 standard deviations from the mean.



14

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

5%

Why would we want to do this? For the answer, we go back to the trade-off between Type I and Type II errors.



15

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

5%

With a 5% test, there is a greater chance of making a Type I error if H0 happens to be true, but there is less risk of making a Type II error if it happens to be false.



16

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

5%

Note that the logic for dropping the left tail depended only on 1 being greater than 0. It did not depend on 1 being any specific value.



17

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

5%

Hence we can generalize the one-sided test to cover the case where the alternative hypothesis is simply that is greater than 0.


alternative hypothesis: H1: > 0

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ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

5%

To justify the use of a one-sided test, all we have to do is to rule out, on the basis of economic theory or previous empirical experience, the possibility that is less than 0.



19

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

5%


alternative hypothesis: H1: < 0

Sometimes, given a null hypothesis H0: = 0, on the basis of economic theory or previous experience, you can rule out the possibility of being greater than 0.

20

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

5%

In this situation you would also perform a one-sided test, now with the left tail being used as the rejection region. With this change, the logic is the same as before.


alternative hypothesis: H1: < 0

We will next investigate how the use of a one-sided test improves the trade-off between the risks of making Type I and Type II errors.

21

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

2.5% 2.5%

Xf

X



22

We will start by returning to the case where can take only two possible values, 0 and 1.

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

2.5% 2.5%

Xf

X



23

Suppose that we use a two-sided 5% significance test. If H0 is true, there is a 5% risk of making a Type I error.

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

2.5% 2.5%

Xf

X



24

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

2.5% 2.5%

Xf

X

However, H0 may be false. In that case the probability of not rejecting it and making a Type II error is given by the blue shaded area.



25

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

2.5% 2.5%

Xf

X

This area gives the probability of the estimate lying within the acceptance region for H0, if H1 is in fact true.



26

Now suppose that you use a one-sided test, taking advantage of the fact that it is irrational to reject H0 if the estimate is in the left tail.

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

5%



27

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

5%

Having expanded the right tail to 5%, we are still performing a 5% significance test, and the risk of making a Type I error is still 5%, if H0 is true.



28

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

5%

But if H0 is false, the risk of making a Type II error is smaller than before. The probability of an estimate lying in the acceptance region for H0 is now given by the red area.



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This is smaller than the blue area for the two-sided test.

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X



30

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

Thus, with no increase in the probability of making a Type I error (if H0 is true), we have reduced the probability of making a Type II error (if H0 is false).



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ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

When the alternative hypothesis is H1: > 0 or H1: < 0, the more general (and more typical) case, we cannot draw this diagram.



32

ONE-SIDED t TESTS

10 0+sd0–sd0–2sd 1+2sd

Xf

X

Nevertheless we can be sure that, by using a one-sided test, we are reducing the probability of making a Type II error, if H0 happens to be false.



33

One-sided tests are often particularly useful where the analysis relates to the evaluation of treatment and effect. Suppose that a number of units of observation receive some type of treatment and Xi is a measure of the effect of the treatment for observation i.

ONE-SIDED t TESTS

Xfnull hypothesis: H0: = 0

alternative hypothesis: H1: ≠ 0

X0 1.96 sd–1.96 sd

2.5% 2.5%

reject H0 reject H0do not reject H0

34

To demonstrate that the treatment did have an effect, we set up the null hypothesis H0: = 0 and see if we can reject it, given the sample average X.

ONE-SIDED t TESTS

Xf

X0 1.96 sd–1.96 sd

2.5% 2.5%

reject H0reject H0 do not reject H0



35

If you use a two-sided 5% significance test, X must be 1.96 standard deviations above or below 0 if you are to reject H0.

ONE-SIDED t TESTS

Xf

X0 1.96 sd–1.96 sd

2.5% 2.5%

reject H0reject H0 do not reject H0



36

ONE-SIDED t TESTS

Xf

X0–1.96 sd

2.5%

do not reject H0

1.65 sd

However, if you can justify the use of a one-sided test, for example with H1: > 0, your estimate has to be only 1.65 standard deviations above 0.

reject H0



37

ONE-SIDED t TESTS

Xf

X0–1.96 sd

2.5%

reject H0

1.65 sd

This makes it easier to reject H0 and thereby demonstrate that the treatment has had a significant effect.

do not reject H0



38

ONE-SIDED t TESTS

Xf

X0–1.96 sd

2.5%

reject H0

1.65 sd

do not reject H0

Throughout this sequence, it has been assumed that the standard deviation of the distribution of b2 is known, and the normal distribution has been used in the diagrams.



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ONE-SIDED t TESTS

Xf

X0–1.96 sd

2.5%

reject H0

1.65 sd

do not reject H0

In practice, of course, the standard deviation has to be estimated as the standard error, and the t distribution is the relevant distribution. However, the logic is exactly the same.



40

ONE-SIDED t TESTS

Xf

X0–1.96 sd

2.5%

reject H0

1.65 sd

do not reject H0

At any given significance level, the critical value of t for a one-sided test is lower than that for a two-sided test.



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ONE-SIDED t TESTS

Xf

X0–1.96 sd

2.5%

reject H0

1.65 sd

do not reject H0

Hence, if H0 is false, the risk of not rejecting it, thereby making a Type II error, is smaller.



Copyright Christopher Dougherty 2012.

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Introduction to Econometrics, fourth edition 2011, Oxford University Press.

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1 11 00 This sequence explains the logic behind a one-sided t test. ONE-SIDED t TESTS 0 +sd 0...

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