Intro to Scientific Analysis (BIO 100) THE t-test. Plant Height (m)

Transcription

1 THE t-test Let Start With a Example Whe coductig experimet, we would like to kow whether a experimetal treatmet had a effect o ome variable. A a imple but itructive example, uppoe we wat to kow whether a ew formulatio of fertilizer icreae plat growth over that of a old fertilizer formula. To tet thi, we might meaure the growth repoe (let ay height) of two et of plat, each of which i grow o oe of the two fertilizer. Let imagie that we grow 0 plat o the old fertilizer ad 0 plat o the ew fertilizer; the height of each idividual plat ad the mea for each fertilizer are give i the table below. Plat Height (m) Old Fertilizer New Fertilizer Mea:.4 Mea:.93 SD: 0. SD: 0.63 A you ca ee, the calculated mea height of plat grow o the ew fertilizer wa greater tha that of plat grow o the old fertilizer. But wait a miute! Before we jump to the cocluio that the ew fertilizer i better tha the old, let take a cloer look at the data that give rie to thee mea plat height. If you look cloely at the data, you ll otice that the data are variable. There variatio i plat height withi each of the fertilizer treatmet, repreeted by the tadard deviatio (SD), ad there alo variatio i plat height betwee the two treatmet. For example, oe of the plat grow o the old fertilizer grew quite tall ad reached.3 m. I fact, thi i taller tha eve out of te of the plat grow o the ew fertilizer ad taller tha the mea height of all plat grow o the ew fertilizer! Thi raie a importat quetio. How ca we claim that the ew fertilizer i i fact better if for ome plat it i ad ome plat it ot? Of coure, a few of the plat grow o the ew fertilizer were taller tha thoe grow o the old fertilizer, but ot all. Correpodigly, may of the plat grow o the old fertilizer were horter tha thoe grow o the ew fertilizer, but ot all. We expect ome variatio. But how much variatio i too much for u to coider that there wa a igificat poitive effect o plat growth of the ew fertilizer? Let look at thee data aother way. Figure how the plat height above a poit o a graph. Fig.. Idividual height (m) meauremet of plat grow i old ad ew fertilizer. The olid lie repreet mea height for each treatmet ( = 0 for each treatmet).

2 Plotted i thi way, you ca ee that the idividual plat height overlap betwee the old ad ew fertilizer; thi i due to radom variatio i plat height withi each treatmet. Although the calculated value of the mea betwee old ad ew fertilizer are differet umber, our cocluio about whether or ot the there wa a effect of the fertilizer treatmet o plat growth overall deped o how much variability there i i the data. The more variable our data, the le cofidet we ca be that the mea reflect a meaigful differece. To drive thi poit home, let examie a dataet of plat height o the ame fertilizer treatmet ad with the ame mea. But thi time the data are le variable. Figure how thi ew dataet a height of idividual plat grow o the old ad ew fertilizer. A a meaure of variability, let ue the tadard deviatio (SD). For the old fertilizer, SD = 0.9; for the ew fertilizer, SD = 0.. Fig.. Idividual height (m) meauremet of plat grow i old or ew fertilizer. Data for each treatmet ha the ame mea a i Fig., but are le variable (SD of old fertilizer = 0.9; SD of ew fertilizer = 0.). If you had a choice betwee uig the data i Figure or Figure to determie whether the old or ew fertilizer differed i their effect o plat height growth, which data would you have the mot cofidece i? Becaue the data i Figure i le variable tha the data i Figure, it tell u that the mea we calculated i Figure are actually more precie tha thoe i Figure. A a reult, we are more cofidet that the mea i Figure differ from oe aother tha we are cofidet that the mea i Figure differ. Thu, the variability of our data i what i truly critical whe makig cocluio about whether or ot real differece actually exit betwee our populatio of iteret. A cietit who are taked with beig objective whe makig uch cocluio, thi i where we tur to tatitical approache. A it tur out, the mea i Figure do ot differ tatitically from oe aother whe they are compared uig a objective tatitical tet, wherea the mea i Figure do differ igificatly. If you were to coclude that imply becaue the calculated mea were differet i the Figure data, the you would have made a icorrect cocluio. Statitic miimize the rik of makig thi type of mitake. The t-tet The t-tet, or Studet t-tet, i a tatitical tet that allow u to compare two ample mea. It i called a t-tet becaue we calculate a tet tatitic called a t-value. The t-value i calculated baed o the differece betwee the two mea but take accout of the variatio i the data. If the differece betwee two mea i large, the it i likely that the two mea are differet. However, a decribed i the fertilizer example above, we mut alo coider the variability i the data. If variatio i the data i low, the it i more likely that ay differece i the mea i ot due to chace aloe but to a factor that i cauig the mea to differ. The ize (or magitude) of the t-value i idicative of how differet our ample mea are with repect to the variace i the data. A large t-value idicate that the ample mea are igificatly differet, wherea a mall t-value idicate o igificat differece betwee the mea.

3 A a example, let compare the deity of a marh gra, called Spartia, betwee two differet marhe. The t-tet provide a ubiaed way of decidig whether ay oberved differece i the mea deity of Spartia betwee the two marhe i real or imply due to chace. A with all tatitical tet, the t-tet tet the ull hypothei. I thi example, the ull hypothei i: Mea deity of Spartia doe ot differ betwee marhe. I other word, our ull hypothei tate that the mea are equal (i.e., x = x ). The t-tet i baed o the t-ditributio, which i a ditributio that give the probability of gettig a particular t- value for a particular igificace level (α) ad degree of freedom (df). Becaue the t-value reflect the magitude of the differece betwee the mea, 0 if the mea are idetical (a rare occurrece). Sice the t-ditributio i baed o the ull hypothei (i.e., that the mea do ot differ), the t- ditributio ha a mea of zero. The greater the differece betwee the mea (aumig low variace), the higher the t-value will be. The higher the t-value, the le probable it i that you could get a t-value that high if the ull hypothei i true (i.e., mea are the ame). The t-ditributio wa developed to take accout of the fact that ample ize () i mall i mot practical applicatio ad, therefore, require a differet ditributio tha the ormal ditributio. Ideed, the t-ditributio i imply a modified form of the ormal ditributio, which if you remember ha certai propertie that allow u to aig probabilitie of occurrece for our data. For example, data poit that fall farther away from the ceter (i.e., the mea) of a ormal curve are le likely (le probable) to occur tha thoe that lad cloer to the ceter of the curve. Thi alo applie to the t-ditributio. The t- tet relie o thi property whe it compare two mea. To ay that two mea are tatitically differet, we ue the % igificace level (P < 0.0). That i, the differece betwee the mea mut be great eough uch that it i improbable that we would get uch a large differece betwee the mea if, i fact, they were the ame. Said aother way, uig the % igificace level, there i a % chace that we would get a t-value outide of the % level if the mea were the ame. The t-ditributio ad it % ( x.%) probability regio are how at right..% probability.% probability The aumptio of the t-tet are: ) the data are ditributed ormally that i, the frequecy ditributio of the data form a ormal (bell-haped) curve; ) the variace of the two ample beig compared are approximately equal; ad 3) the ample are idepedet. I a t-tet, the t-value i calculated baed o the differece i the ample mea ( x ad x ) ad the tadard error of the differece betwee ample mea ( ): x x 0 t-ditributio It tur out that: x x x x x x = + 3

4 Thi i becaue there i a mathematical relatiohip that equate the tadard error of the differece i the mea with the um of the tadard error of the two variable. Subtitutig thi for i the equatio for t, we get: x x x x + where the i value are the variace of the idividual variable. Thi equatio ca be rearraged a follow: x x + where equal the pooled tadard deviatio of both ample: = i= x i i= ( xi ) + xi i= + i= ( x ) i For ay particular t-tet, the calculated t-value will lie omewhere withi the t-ditributio. The t- value we get will have a probability aociated with it. That probability will be a meaure of how likely it would be for u to get a t-value a large or larger tha we did by chace aloe. Therefore, if our t-value i large (toward the tail of the curve), the there i a low probability that the mea are equal. By cotrat if t i mall (toward the ceter of the curve, that i zero), the the probability that the ull hypothei i true i high. By covetio, we et the probability cut-off poit for the t-value at 0.0 (or %) o the curve. Thi cut-off value of t i called the critical value. The critical t-value for ay particular t-tet ca be looked up i a t-table (called Critical value of the t ditributio ; icluded below). To determie the critical t value you eed to kow the igificace level (α) (i thi cae % or 0.0) ad the degree of freedom (df). The df for a t-tet i: df = ( + - ). If the t-value calculated from a t-tet i larger tha the critical value, the we reject the ull hypothei. By cotrat, if the t-value calculated from a t-tet i maller tha the critical value, the we fail to reject the ull hypothei. 4

5 Example of a t-tet For the Spartia example above, let ay that we collect plat cout from five m plot i each of the two marhe. We get the followig data: Plot # Marh A Marh B x 9.9 x 3.6 x 99.6 x 66.3 x x Uig our calculatio equatio for the pooled tadard deviatio (): + = ( 99.6) ( 66.3) Therefore: = ad which give, = = ( 0.4) = 8. The critical t-value at the 0.0 igificace level with 8 degree of freedom (df) i.306 (i.e., t (0.0, 8) =.306). Becaue our calculated t-value ( 8.) i higher tha the critical t-value at the 0.0 igificace level, there i le tha a % (P < 0.0) chace that we could get a t-value a large or larger tha we oberved if the ull hypothei i true. Therefore, we coclude that the mea deity of Spartia i marh A i igificatly differet tha it deity i marh B. Whe evaluatig tatitical igificace, you mut alway report the igificace level ued ad the P-value i your reult.

6 CRITICAL VALUES OF THE t DISTRIBUTION Sigificace level (α -tailed) Sigificace level (α -tailed) df df