Recall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and nonusers, x  y.


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1 Testig Statistical Hypotheses Recall the study where we estimated the differece betwee mea systolic blood pressure levels of users of oral cotraceptives ad ousers, x  y. Such studies are sometimes viewed as attempts to aswer a questio like "Does the mea blood pressure of OC users differ from that of ousers? (Are x ad y differet?)" This speculatio, that maybe there really is o differece (maybe oral cotraceptive use has o effect o blood pressure), is a hypothesis H : X Y or H : X Y The alterative to H is aother hypothesis, H1 : X Y or H1 : X Y The alterative hypothesis says that the mea blood pressure of OC users really is differet from that of ousers. (It does't say which mea is bigger, or how big the differece is, oly that there is a differece.) Authors: Blume, Greevy Bios 311 Lecture Notes Page 1 of 18
2 I this example, the parameter of iterest, x  y, has a wide rage of possible values, ad x  y =, is special because it represets "o differece" (betwee OC users ad ousers), or "o effect" (of OC use). H is called the ull hypothesis. It states that the parameter x  y equals the special value, zero. H 1, the alterative hypothesis, cosists of the complemet of H. It states that the parameter does ot equal zero. I our example we had observatios o 8 OC users ad 21 ousers. The immediate questio is What do these observatios tell us?" or Is it right to say that these data represet strog evidece that there is ideed a differece betwee the mea blood pressure levels of OC users ad ousers? or Do these observatios justify rejectig H? Authors: Blume, Greevy Bios 311 Lecture Notes Page 2 of 18
3 (Note that: we caot directly aswer the questio Which of these hypotheses is true?" because we oly have a sample of the populatio to work with.) We could aswer the previous questios usig our ew cofidece iterval tools: Costruct a 95% cofidece iterval for x  y, ad aswer "Yes, our data show that there is a differece" if the value x  y = is ot i the iterval. That is, reject the hypothesis of o differece (H ) if the value x  y = is ot i the cofidece iterval. But if the value x  y = is iside the 95% CI, our aswer is "These observatios do ot allow us to reject H." We are ot sayig that the observatios are evidece supportig H. (Remember that our best estimate of x  y was 5.42, ot.) But the data do ot represet strog eough evidece agaist H to justify its rejectio. Authors: Blume, Greevy Bios 311 Lecture Notes Page 3 of 18
4 This procedure is essetially a classical hypothesis test: Whe H specifies the value of a parameter, (as it does i our example) (i) Costruct a 1(1)% CI for the parameter. (ii) Reject H if the hypothesized value is ot i the iterval. Recall our earlier iterpretatio of the CI as: "the values of the parameter that are cosistet with the observatios." What we're doig here is checkig to see if the value specified by H is "cosistet with the observatios". If it is ot, the we reject H. A more mathematically direct approach has prove popular may areas of sciece. Let s look at hypothesis testig more formally. Authors: Blume, Greevy Bios 311 Lecture Notes Page 4 of 18
5 A More Direct Approach To Hypothesis Testig I the blood pressure example, if the two variaces are assumed to be equal the T* X Y S m 2 P m2 X 1 1 m Y ~ t Uder our Null hypothesis (the mea pressures of OC users ad ousers are o differet), H : x  y =, m=21, =8 so T* ~ t SP 8 21 has Studet's t distributio with 27 df (uder H ). For a observed sample we ca calculate the value of this statistic. (T * is called the test statistic ) If it falls i oe of the tails of the tdistributio (say either T * > 2.52 or T * < 2.52) the we reject H. Authors: Blume, Greevy Bios 311 Lecture Notes Page 5 of 18
6 A popular explaatio for the reasoig behid this procedure is as follows: If H is true the T * will usually (95% of the time) fall betwee ad If it falls outside this iterval, the either (a) H is true, ad we just happeed to observe a relatively rare sample where T * falls this far from its expected value,, or (b) H is false; the expected value of T * is really ot. Values of T * that are far from zero (large values of T * ) are improbable uder H. If we observe such a value, we reject H. I this example we chose 2.52 as the critical value for our test, ad used the rule "Reject H if T * exceeds 2.52" This rule will sometimes lead us to reject H whe it is i fact true. It is easy to calculate the probability of makig this error: P * * T 2.52 H 1 P 2.52 T Authors: Blume, Greevy Bios 311 Lecture Notes Page 6 of 18
7 By choosig this critical value, we cotrol the error probability. If we use a larger critical value, say t = 2.771, the we reduce the probability that we will reject H whe it is really true: P * * T H 1 P T This is a example of aother geeral scheme for hypothesis testig: Istead of costructig a cofidece iterval, choose a observable "test statistic" (T * ) for which (i) the probability distributio is kow (at least approximately) whe H is true, ad (ii) the larger the value of the test statistic, the stroger the evidece agaist H. The we choose some critical value, ad reject H if the observed value of the test statistic exceeds the critical value. Authors: Blume, Greevy Bios 311 Lecture Notes Page 7 of 18
8 I our OC example the test statistic was T * ~ t SP 8 21 Whe H is true, T * has a Studet's t distributio with 27 df. Sice the distributio of the test statistic whe H is true is kow, we ca choose the critical value to fix the error probability (reject H whe H is really true) at whatever value we like. Popular choices are.5 ad.1 (1/2 ad 1/1). This error probability is ofte called the "sigificace level" of the test ad ofte represet by =P(reject H H true) (Remember, we use the fact that H is true to costruct the test statistic.) Authors: Blume, Greevy Bios 311 Lecture Notes Page 8 of 18
9 Testig a Hypothesis About a Sigle Mea Assume that the data have ormal probability distributio, variace kow. The Null hypothesis is H : =. The Test statistic: T * X Notice the form of T *? * X T E X H Var X This is a appropriate test statistic for this hypothesis, because (i) its distributio whe H is true is kow (stadard ormal), ad (ii) the larger the value of the test statistic, the farther the sample mea, X, is from, ad it seems appropriate to say that the farther X is from, the stroger the evidece that is ot the expected value of X. Authors: Blume, Greevy Bios 311 Lecture Notes Page 9 of 18
10 If we choose the critical value 1.96, the we have P reject H X H P 1.96 H P Z This gives a test of H at the 5% sigificace level. To test H at the 1% sigificace level, we must use a larger critical value, P reject H X H P H P Z To reject H at the 1% level requires stroger evidece (a more extreme value of the test statistic) tha is required to reject at the 5% level. At the 5% level we reject the hypothesis that H : =E[X]= if X falls further tha 1.96 SE's away from the hypothesized value,. At the 1% level we oly reject if X falls further tha SE's away from. Authors: Blume, Greevy Bios 311 Lecture Notes Page 1 of 18
11 Example We kow (Pagao ad Gauvreau) that the distributio of serum cholesterol i adult males i the U.S. has a mea of 211 mg/1ml ad a stadard deviatio of mg/1ml. We wat to lear about me who are hypertesive smokers. Specifically, is the distributio of serum cholesterol amog such me ay differet from that i the geeral populatio? Let's suppose the stadard deviatio is the same as i the geeral populatio ( mg/1ml), ad test whether the mea, μ, is the same or ot. H : = 211 mg/1ml H 1 : 211 mg/1ml Our test statistic will be T * X 211 which, uder H, has a stadard ormal distributio. Authors: Blume, Greevy Bios 311 Lecture Notes Page 11 of 18
12 Thus to test at the 5% level, we will reject H if the absolute value of the test statistic exceeds This meas that we will reject uless X which implies that we will reject uless the 95% CI cotais the hypothesized value, 211: X X 1.96??? Authors: Blume, Greevy Bios 311 Lecture Notes Page 12 of 18
13 I Pagao's example, the sample size is = 12 ad the sample mea is 217, so the test statistic's value is T *.45 Sice this is much less tha the critical value, 1.96, we do ot come close to rejectig H. (The 95% CI is (191, 243), ad the hypothesized value, 211, is well withi the iterval.) If we take a much larger sample of hypertesive smokers, say 25 istead of 12, ad fid the same value for the sample mea, X 25 = 217, the test statistic will be T * 2.6 Sice this is greater tha the critical value, 1.96, the result would ow be "Reject H at the 5% level." Authors: Blume, Greevy Bios 311 Lecture Notes Page 13 of 18
14 Whe the observatios lead to rejectio of H at a specified level, such as 5%, they are said to be "statistically sigificat at the 5% level." (Whe they do't, they are "ot statistically sigificat.") We saw that a sample of size 12 with mea 217 is ot statistically sigificat for testig H : =211 at the 5% level ( σ = kow ). A sample of 25 with the same mea, 217, is statistically sigificat at the 5% level (but ot at the 1% level, sice the test statistic's value, 2.6, is less tha the 1% critical value, ) The 95% CI is 217 ± 1.96(/ 25 ), or ( 211.3, ), which excludes the hypothesized value, 211, ad The 99% CI is 217 ± 2.576(/ 25 ), or ( 29.5, ) which icludes that value. Authors: Blume, Greevy Bios 311 Lecture Notes Page 14 of 18
15 Textbooks ofte stress the importace of selectig the sigificace level (choosig the value for the error probability, α) before `lookig at the data. We will see later why they stress this. I practice it is very commo ot to select a fixed α at all. Istead, what is ofte doe is that for the value of the test statistic that is actually observed, the probability of observig a value that large or larger (uder H ) is calculated ad reported. This quatity is called the pvalue. It is a measure of how extreme the test statistic is, ad it is used as a measure of how strog the evidece agaist H is. We've bee cosiderig a test procedure that chooses a test statistic, T *, ad a fixed error probability,. The it fids the critical value, say t, for which P(T * > t H )=. This procedure rejects H if the observed value of the test statistic, say T obs *, exceeds the critical value, t. This is equivalet to fidig the pvalue, P(T * > T obs * H ), ad the rejectig H if this probability is less tha. Authors: Blume, Greevy Bios 311 Lecture Notes Page 15 of 18
16 For the hypertesive smokers' cholesterol levels (whe = 25) the test statistic was T * 25 X ad the observed value of the test statistic was T * obs The probability (uder H ) of observig a value of the test statistic this extreme is 25 X 25 P(T * > T * obs H )= P 2.6 H PZ Sice this is greater tha.1, but smaller tha.5, it shows that our observatios represet evidece strog eough to justify rejectio at the 5% sigificace level, but ot strog eough for the 1% level. Authors: Blume, Greevy Bios 311 Lecture Notes Page 16 of 18
17 A ote o the Pvalue The pvalue P(T * > T obs * H ), is somethig very familiar. It is just a probability statemet about the sample average. P(T * > T * obs H )= P P X P X X H Which says that the probability of observig a sample mea greater tha or equal to the 217 that was observed is oly.39 if the true mea is really 211 (the ull hypothesis is true). Authors: Blume, Greevy Bios 311 Lecture Notes Page 17 of 18
18 Two Types of Errors We have see that for testig the hypothesis that the mea of a ormal probability distributio has a specified value, H : = (agaist the alterative hypothesis that the mea is ot ), we reject H at sigificace level =.5 if T * X 1.96 This procedure ca lead us to reject the hypothesis H whe it is really true. The probability of this error is.5, ad is ofte called the Type I error, sigificace level, or the size of the test. But what if the hypothesis H is false ad we fail to reject H? This error is called the Type II error ad represet by the symbol. Type I error : =P(reject H H true) Type II error: =P(fail to reject H H false) Power: 1= P(reject H H false) (Loose defiitio we ll be more specific later) Authors: Blume, Greevy Bios 311 Lecture Notes Page 18 of 18
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