1 Statistics. We ll examine two ways to examine the relationship between two variables correlation and regression. They re conceptually very similar.

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1 1 Statistics Rudolf N. Cadinal NST IB Psychology Pactical 1 (Tue 11 & Wed 1 Novembe 003). Coelation and egession Objectives We ll examine two ways to examine the elationship between two vaiables coelation and egession. They e conceptually vey simila. Stuff with a solid edge, like this, is impotant. But emembe you can totally ignoe stuff with single/double wavy bodes..1 Scatte plots Suppose you measue two things about a goup of subjects IQ and income, say. How can we establish if thee s any elationship between the two? The fist thing to do is to daw a scatte plot of the two vaiables. To do this, we take one of ou vaiables (e.g. IQ) as the x axis, and the othe as the y axis. Each subject is then plotted as one point, epesenting an {IQ, income} pai. This might show us any of seveal things: Fictional scatteplots. A: positive coelation between IQ and income. As IQ goes up, income goes up. B: negative coelation between IQ and income. As IQ goes up, income goes down. C: no coelation between the two. D: thee s a elationship, but it s not a staight line (it s not a linea elationship). People with high IQs and people with low IQs both ean less than those with middling IQs. It s always woth plotting the data like this fist. Howeve, fo ou next tick we d like a statistical way to wok out if thee s a elationship, how big it is, and in what diection it goes. Please note that we ll only talk about ways to establish things about a linea elationship between two vaiables; if it s non-linea (e.g. the bottom ight figue), it s beyond the scope of this couse.. Coelation We will call the degee to which they ae elated the coelation between the two vaiables. If gets bigge when X gets bigge, thee s a positive coelation; if gets smalle when X gets bigge, thee s a negative coelation; if thee s no linea elationship, thee s a zeo coelation. Hee s how we wok it out.

2 The covaiance Fist, we need some sot of numbe that tells us how much ou two vaiables vay togethe. Let s suppose we have n obsevations. Let s call ou two vaiables X and. We fist find the two means, x and y. Then we can calculate something called the sample covaiance: ( x x)( y y) cov X n 1 Look at the fist pat of the equation fist it s vey like the sample vaiance (if we changed all the ys to xs in this equation, we d have s X ; if we changed all the xs to ys, we d have s ). (And yes, if you e wondeing, if we wanted the population covaiance, we d divide by n athe than n 1, but we don t.) Pehaps you can see fom the equation how it woks. Fo a given {x, y} point, if x is vey fa above the x-mean ( x ), and y is vey fa above the y-mean ( y ), then a big numbe gets added to ou covaiance. Similaly, if x is vey fa below the x-mean ( x ), and y is vey fa below the y-mean ( y ), then a big numbe gets added to ou covaiance. Both these occuences suggest a positive linea elationship (like the top-left pat of ou figue). On the othe hand, if x is vey fa above x, and y is vey fa below y, then a lage negative numbe gets added to ou covaiance; the same s tue if x is vey fa below x and y is vey fa above y. Points nea the mean don t tell us so much about the elationship between x and y, and they don t contibute much to the covaiance scoe. If thee s no elationship between X and, then when x is above x, about half the time y will be above y and the covaiance will get bigge, but about half the time y will be below y and the covaiance will get smalle, so the covaiance ends up being about zeo. The Peason poduct moment coelation coefficient, The covaiance tells us how much the two vaiables ae elated, but it has a poblem the actual value of the covaiance depends on the standad deviations of ou two vaiables as well as the coelation between them. A covaiance of 140 might be an high coelation if the standad deviations ae small, but a poo coelation if the standad deviations ae lage. We can get ound this poblem by calculating : cov X X s s It tuns out that vaies fom 1 (pefect negative coelation), though 0 (no coelation), to +1 (pefect positive coelation). Incidentally, the coelations in ou pictue wee (figue A), 0.81 (figue B), 0.10 (figue C), 0.04 (figue D). Zeo coelation doesn t imply no elationship That should be immediately appaent: I ve just told you that the coelation between IQ and income in figue D was 0.04, nealy zeo, and yet thee s clealy a vey stong elationship it just isn t a linea one. Always plot you data to avoid dawing mistaken conclusions fom values. Coelation does not imply causation Finding that X and ae elated does not mean that X and ae causally elated. It s easy to jump to this assumption if the elationship is plausible we might intuitively think that cleve people get bette jobs, fo example, and thus accept a positive coelation between IQ as income as indicating causation. It doesn t. Maybe the causal elationship is backwads: having moe money might impove you IQ. Maybe the two ae connected though a thid vaiable: Z causes X and Z causes (e.g. maybe having ich paents means you e moe likely to have a high IQ, and X

3 3 also makes you moe likely to get a well-paid job as an adult). The point is, we just can t tell fom the plain coelation. Adjusted If we measued IQ and income fo a sample of just two people let s say {IQ 110, 0,000} and {IQ 10, 5,000} and calculate, we ll find that thee s a pefect coelation, +1. If you plot only two points on a scatteplot, you can always join them pefectly with a staight line. This doesn t mean that the coelation is +1 in the population! So thee s something slightly wong with ou sample coelation statistic, it s a biased estimato of the population coelation, which we wite as ρ (Geek lette ho). We can to do something to make it a bette (unbiased) estimato. We can calculate the adjusted, adj : adj (1 )( n 1) n If the sample size is lage, and adj will be about the same. Please note that doing this will give you a positive value fo adj, since squae oots can t be negative so you need to look at the oiginal data o value to wok out which way (+ o ) the coelation should be. Bewae if you coelation is based on a esticted ange of data It s obvious that if we sample too few data points, we won t get a vey good estimate of fo ou population (that s what calculating adj is meant to sot out). But it should also be clea that if we sample fom a esticted ange, we can also get the wong answe, even if we sample many obsevations within that esticted ange. Hee s an exteme example (figues E G below): depending on the ange of data we sample, we can contive to find a negative, zeo, o positive coelation between ou two vaiables. Bewae outlies Above: Sampling a esticted ange of data can oveestimate (E) o undeestimate (G) compaed to sampling the whole ange (F). Black dots ae pat of the sample; white dots ae pat of the population that wasn t sampled. The staight line epesents the coelation. Below: Outlies can have lage effects on. In (H) the outlie makes nealy 0; without it, would be nealy 1. In (I), the outlie makes nealy 1; without it, would be nealy 0.

4 4 Exteme values, o outlies, can have lage effects on the coelation coefficient. (We won t talk about what to do with them in the IB couse, but you should be awae of the poblems they can cause.) Two examples ae shown in figues H I. Bewae if you population has distinct subgoups We can also encounte poblems if ou measuements aen t fom one homogeneous population. A couple of examples ae shown in figues J K (but subgoup effects can be a good deal moe subtle than this!). Fictional data illustating poblems with subgoups. (J) Coelation between height and weight fo vaious things we found in a magic foest. If we measue an oveall coelation, we may find that tall things weigh less (negative coelation between height and mass), but this is only because we have two vey diffeent subgoups. But we have heteogeneous subsamples within each subgoup (wild boa and unne beans) thee is a positive coelation. (K) A less stupid example. If we ae investigating whethe something is cacinogenic, we might find a negative coelation, suggesting (if we have designed ou expeiment so that we know that the chemical caused any obseved change in cance ate) that the chemical potects fom cance. But we must check, because this could be due to a subgoup effect: a moe detailed analysis may eveal a vulneable subgoup (who get high ates of cance) and a esistant subgoup (who aen t as likely to get cance); in this example, the ates of cance ae actually inceased by the chemical in both subgoups..3 Is a coelation significant? Assumptions we must make If all we want to do is to descibe the sample that we have (e.g. with coelation and/o egession), we don t have to make any assumptions although coelation and egession both aim to descibe a linea elationship between two vaiables, so if the elationship isn t linea, then the answes we get fom coelation and egession won t mean vey much. But if we want to pefom statistical tests with the data (e.g. is this coelation coefficient significantly diffeent fom zeo? ), we will effectively be asking questions to do with the undelying population that ou sample was dawn fom (i.e. what is the chance that a sample with coelation came fom an undelying population with coelation ρ 0? ). This equies making some assumptions, o ou statistical tests won t be meaningful. Basically, the data shouldn t look too weid: The vaiance of should be oughly the same fo all values of X. This is often called homogeneity of vaiance; its opposite, what you don t want, is called heteoscedasticity (Geek homo same, heteo othe, skedastos able to be scatteed). Heteoscedasticity: a Bad Thing. The vaiance in income is vey diffeent fo low-iq and high-iq data.

5 5 If we ae asking questions about ρ, we must assume that both X and ae nomally distibuted. Fo all values of X, the coesponding values of should be nomally distibuted (and vice vesa). [ou may see the last two assumptions efeed to togethe as the assumption of bivaiate nomality.] Testing the significance of is significantly diffeent fom zeo? Let s suppose we take a sample of people, measue thei IQs and incomes, and coelate them to find. That s the coelation in the sample; but is thee a coelation in the whole population? Ou null hypothesis is that the population coelation coefficient (ρ) is zeo. Without going into the details, we can compute a numbe called a t statistic: t n n We can use this numbe, t, to pefom a t test with n degees of feedom. (The statistical catchphase is that the numbe we have just calculated is distibuted as t with n degees of feedom ; the t distibution is much like the nomal distibution that we ve mentioned befoe, so we need to look up the pobability coesponding to ou t statistic just like we might look up a pobability coesponding to a Z scoe.) To intepet this using tables, we can look up the citical value of t fo ou paticula value of α and the numbe of degees of feedom; if ou t statistic is bigge than the citical value, it s significant and we eject the null hypothesis that thee s no coelation in ou undelying population. (Note that we use, not adj, fo this test.) This is an example of a t test; we ll cove these popely in Pactical..4 Speaman s coelation coefficient fo anked data ( s ) If ou X and data ae both anked (see below fo how to ank data), we can calculate the coelation coefficient just as nomal, except that we ll call it s (sometimes called Speaman s ho). Howeve, when we want to test the significance of s, we have a poblem, because we cannot make ou assumption that the data ae nomally distibuted. Some ague that thee ae substantial poblems inheent in computing the significance of s (see Howell, 1997, p. 90). Anyway, with these caveats, what we ll do is to look up citical values of s if n 30, and if n > 30 we ll calculate t and test that, just as befoe: s n tn whee n > 30 s To answe the question ae these values in a paticula ode? you can coelate the ank of the data with the ank of thei position. Fo example, suppose you take lage spoonfuls of ban flakes fom the top of a ceeal packet, one by one, and find the mean weight of individual ban flakes in each spoonful. These weights, in milligams and in ode, ae 70, 84, 45, 50, 48, 40, 38, 40, 5, 30. If you want to establish whethe it s tue that big ban flakes come out of the packet fist, you can coelate the set of positional anks {1,, 3, 4, 5, 6, 7, 8, 9, 10} with the coesponding anks of the data {9, 10, 6, 8, 7, 4.5, 3, 4.5, 1, } to get s (p <.001). How to ank data Suppose we have ten measuements (e.g. test scoes) and want to ank them. Fist, place them in ascending numeical ode:

6 6 Then stat assigning them anks. When you come to a tie, give each value the mean of the anks they e tied fo fo example, the 1s ae tied fo anks 4 and 5, so they get the ank 4.5; the 16s ae tied fo anks 7, 8, and 9, so they get the ank 8: X: ank: Regession We ve used coelation to measue how much of a elationship thee is between two vaiables. We can use a elated technique, egession, to establish exactly what that elationship is specifically, to make pedictions about one vaiable using the othe. Suppose thee s a positive coelation between seum cholesteol in 50-yeaold men and thei chance of having a heat attack in the next five yeas. If M Blobby has a seum cholesteol twice that of M Slim, ae his chances of having a heat attack doubled? Inceased by a facto of 1.5? Tipled? Let s find out. If we call ou two vaiables X (cholesteol) and (chance of having a heat attack), we can wite an egession equation that descibes the linea elationship between X and. It s just the equation of a staight line: ˆ bx + a We call this the egession of on X, meaning that we e pedicting fom X, not the evese. The with a hat (Ŷ ) just means the pedicted value of. This is the pictue that this equation epesents: The egession equation and what it means. ou might also see it witten y y 0 + ax, o some othe equivalent. We could daw thousands of lines like this. So which one is the best fit to ou data? If we take a paticula line ˆ bx + a, then fo each {x, y} point, we can calulate a pedicted value y ˆ bx + a. Fom this, we can calculate how wong ou pediction was: the pediction eo is y yˆ. This eo is often called the esidual, because it s what you have left afte you ve made you pediction. Since this will sometimes be positive and sometimes be negative, we can squae it to get id of the +/ sign, giving us the squaed eo: ( y yˆ). So we should aim to find a line that gives us the minimum possible total pediction eo, o sum squaed eo, ( y yˆ). (This pocedue is called least squaes egession.) As it happens, this is when a y bx cov b s X X s (o b s X if that s easie with you calculato) Note that egession is not a symmetical pocess: the best-fit line fo pedicting fom X is pobably not the same as the best-fit line fo pedicting X fom (illustated in the figue below). This is diffeent fom coelation, which doesn t cae which way ound X and ae. To save you the bothe of doing this by hand, you calculato should give you A and B (egession) and (coelation) diectly. Lean how to use it fo the exam! Typically, you put it into statistics mode o linea egession mode, clea the stats

7 7 Residuals and lines of best fit. (A) What s a esidual? (B D), which ae LESS IMPORTANT, show why pedicting fom X is diffeent fom pedicting X fom. memoy, then ente each data point as an {x, y} pai then you can ead out the answes. Plotting and intepeting the egession line To plot the line, you just need any two {x, ŷ } pais though it helps if they e fa apat, because this makes you line moe accuate, and it s often wise to plot a thid point somewhee in the middle to make sue it lies on the same line! The line will also pass though the points {0, a} and { x, y }. If you actually need to pedict a y value fom some x value say you fathe s got a paticula cholesteol level and you wanted to pedict his isk of a heat attack then you can just use the egession equation diectly. Bewae of extapolating beyond the oiginal data, though. If you ve based you egession equation on 50- yea-old men with a cholesteol level of 4 8 mm, they may be petty useless at pedicting heat attack isks in 50-yea-old men with a cholesteol level of 1 mm, o 100-yea-old men, o 50-yea-old women. Within the ange of you data, though, you can also make statements like fo evey 1 mm dop in cholesteol, one would expect a 10% eduction in the isk of a heat attack (o whateve it is); this infomation is based on the slope of the egession line. Finally, emembe that coelation and egession do not necessaily epesent causation (see above). as a measue of how good a coelation o egession is So fa, we ve dawn a egession line. But how good it is at pedicting fom X depends on how much of a elationship thee is between and X we could daw a

8 8 egession line whee was the chance of having a heat attack and X was shoe size, but it wouldn t be a vey good one. How can we quantify how good ou best fit is? epesents the popotion of the vaiability in that s pedictable fom the vaiability in X, o (equivalently) the popotion by which the eo in you pediction would be educed if you used X as a pedicto. Let s say the coelation between cholesteol levels and heat attack isk wee idiculously high, at 0.8; then % of the vaiability in the isk of heat attacks would be attibutable to vaiations in cholesteol. If 0.1, then % of the vaiability in the isks of heat attacks would be attibutable to diffeences in cholesteol levels. Note, once again, that this doesn t tell you anything about causality. If ainfall is pedictable fom twinges in you gammy knee, that doesn t necessaily mean that twinges cause ain, o that ain causes twinges. Two egessions with nealy identical equations ( ˆ X ) but diffeent values of. Mathematical statement of this popety of Let s stat by taking the wost-case scenaio. If you knew nothing about you subject s cholesteol level (X), how accuately could you pedict his isk of a heat attack ()? ou best guess would be the mean isk of a heat attack, y, and you eo would be descibed in some way by the standad deviation of, s, o the vaiance s. The vaiance, emembe, is ( y y) s n 1 Now the bottom pat of that, n 1, is the numbe of degees of feedom (df) ou estimate of the vaiance was based on. (This was in the fist handout if you have n numbes, and you use them to calculate the sample mean, y, then you can subsequently only alte n 1 of the numbes feely without alteing the mean. This is called the numbe of degees of feedom you have left it is the numbe of independent obsevations on which a given estimate is based.) The top pat is the sum of the squaes of the deviations of fom the mean of, which we shoten to the sum of squaes of (SS ). So we can wite the vaiance as SS s df Let s now suppose that we do know ou subject s cholesteol. We have a whole set of n obsevations with which to calculate a egession line, i.e. a and b. (Since we calculate two numbes, we e left with n degees of feedom in ou data.) But now we can estimate ou subject s heat attack isk athe bette, we hope and the eo in doing so will be elated to the esiduals (eo) of ou egession s pediction. This thing is called the esidual vaiance, o eo vaiance: ( y yˆ) SSesidual sesidual n df

9 9 and its squae oot, s esidual (sometimes witten s X to show that has been pedicted fom X), is called the standad eo of the estimate (it s like a standad deviation the squae oot of the vaiance of the eos is the standad deviation of the eos, abbeviated to the standad eo). Well, athe than wok all this out by hand, let s use somebody else s esult: n 1 sesidual s ( ) s adj n Actually, it s geneally easiest to do the calculations in tems of the sums of squaes, not vaiances, because then we don t have to woy about all these degee-offeedom coections ( and adj and this n 1, n business) you can t add two vaiances togethe unless they e based on the same numbe of degees of feedom, but you can add sums of squaes togethe any way you like and we find that SS SS (1 ) esidual SS ( SS SS esidual SS SS ) + SS esidual In othe wods, the total vaiability in is made up of a component that s elated to X ( SS SS SS esidual, which we can also wite as SS ˆ, the vaiability in the pedicted value of ) and a component that s esidual eo (SS esidual ). Tanslated to ou cholesteol example, people vay in thei cholesteol levels (SS X ), they vay in thei heat attack isk (SS ), a cetain amount of the vaiability in thei heat attack isk is pedictable fom thei cholesteol ( SS ˆ ), and a cetain amount of vaiability is left ove afte you ve made that pediction (SS esidual ). O, SS S + SS whee.6 Advanced eal-wold topics ˆ SS SS ˆ esidual As with all the wavy-line sections, this section cetainly isn t intended to be leaned! It s fo use with eal-wold poblems that you may encounte. ou will not be tested on any of this in the exam. What s egession to the mean? Something elated to egession, but quite inteesting. It was discoveed by Galton in He measued the heights of lots of families, and calculated the mid-paent height (the aveage of the mothe s and the fathe s height) call it X and the heights of thei adult childen call it. He found that the aveage mid-paent height was x 68. inches; so was the aveage height of the childen ( y 68. inches). Now, conside those paents with a mid-paent height of inches; the mean height of thei childen was 69.5 inches the height of these childen (69.5) was close to the mean of all the childen ( y 68.) than the height of the paents (70 71) was to the mean of all the paents ( x 68.). But this wasn t a genetic phenomenon, it was a statistical phenomenon, and it woked backwads: if you took childen with a height of inches, the mean mid-paent height of thei paents was 69.0 inches. This is called egession to the mean. Why does it happen? Suppose we have the vaiables X and, with standad deviations s X and s, and the coelation between them is. We ve peviously seen that cov X X s X s and the egession slope b is

10 10 Theefoe, cov b s s slope b s X So a change of one standad deviation in X is associated with a change of standad deviations in. And we know the egession line always goes though the point at the means of both X and that is, the point { x, y }. Theefoe, unless thee is pefect coelation ( 1), the pedicted value of is always fewe standad deviations fom its mean than X is fom its mean. Remembe that pedicting fom X is diffeent fom pedicting X fom, unless the two ae pefectly coelated? This is anothe way of saying the same thing. Examples of egession to the mean (fom Bland & Altman, 1994, BMJ 309: 780) If we ae tying to teat high blood pessue, we might measue blood pessue at time 1, then teated, and then measued again at time. We might see that blood pessue goes down most in those who had the highest blood pessue at time 1, and we might intepet this as an effect of the teatment. We d be wong; this is egession to the mean. It would happen even if the teatment had no effect. The two sets of obsevations (time 1, time ) will neve be pefectly coelated (because of measuement eo and biological vaiation); < 1. So if the diffeence between ou high blood pessue subgoup and the whole population was q at time 1, it will be q at time i.e. the diffeence fom the population mean will have shunk. We should have compaed ou teated goup to a andomized contol goup. In one study, people epoted thei own weight and had thei weight measued objectively. A egession was used to pedict epoted weight fom measued weight; the egession slope was less than 1. This might lead to you intepet that vey fat people undeestimate thei weight when they epot it, and vey thin people oveestimate it. But we d neve have expected pefect coelation. All this might be is egession to the mean and if we d pedicted measued weight fom epoted weight, we d also have a slope less than one, fom which we might have concluded the opposite: that vey fat people oveestimated thei weights and vey thin people undeestimated them. When scientific papes ae submitted to jounals, efeees citicize them and editos select the best ones to publish on the basis of the efeees epots. Because efeees judgements always contain some eo, they cannot be pefectly coelated with any measue of the tue quality of the pape. Theefoe, because of egession to the mean, the aveage quality of the papes that the edito accepts will be less than he thinks, and the aveage quality of those ejected will be highe than he thinks. X X Patial coelation dealing with the effects of a thid vaiable Sometimes we ae inteested in the elationship between two vaiables and know that a thid vaiable is also influencing the situation. Imagine we examine the coelation between IQ (X) and income (), and find it to be positive, but we suspect that one eason that highe IQ pedicts highe income is because people with highe IQs ae moe likely to get into univesity, stay fo highe degees, and so on and it s the degee that gets you the highe income, not you IQ itself. So is thee any futhe elationship between IQ and income once you ve taken into account this effect of studying fo longe? One way of investigating this is to look at the coelation between IQ (X) and (say) numbe of yeas of study (Z), and the coelation between income () and numbe of yeas of study (Z). We can then calculate the patial coelation between IQ (X) and income () having taken account of the elationship of each of these to numbe of yeas of study. We call this patialling out the effects of numbe of yeas of study. We tem the patial coelation coefficient between X and with the effects of Z patialled out xy.z, and calculate it like this:

11 11 xy. z xy (1 xz xz yz )(1 yz ) Let s use some fictional numbes to illustate this: suppose that the coelation between IQ and income is xy 0.6, the coelation between IQ and yeas of study is xz 0.8, and the coelation between income and yeas of study is yz 0.7. Then the coelation between IQ and income having patialled out the effect of yeas of study would be only xy.z This would mean that xy.z , so only 0.8% of the vaiability in income is pedictable fom IQ once you ve taken account of the numbe of yeas of study, even though xy % of the vaiability in income is pedictable fom IQ. This would suggest that nealy all the ability of IQ to pedict income was due to the fact that high IQs pedict moe yeas of study. The point-biseial coelation ( pb ) fo a dichotomous vaiable If we ask the question is body weight coelated with sex?, we have a bit of a poblem with assuming that sex is nomally distibuted; it clealy isn t. Body weight pobably is, but mammals ae eithe male o female; the sex vaiable is dichotomous (Geek dikhotomos, fom dikho-, in two; temnein, to cut). No poblem: simply assign two values to the dichotomous vaiable as you see fit e.g. male 0, female 1 (o male 56, female 98; it doesn t matte at all). Then calculate as nomal. Officially this is called pb, the point-biseial coelation coefficient, but you can teat it like any, and test it fo significance in the same way (a t test on n df) as we saw befoe: t n n ou might think that asking does weight vay with sex and calculating a coelation is a bit daft hee, and the moe natual question is do the sexes diffe in body weight? ou d be ight, eally, but it is actually the same question. Since it s the same question, thee must be a simple elationship between pb and the t statistic: t pb t + df The use of this is that if you test the diffeence between two goups (e.g. male body weights and female body weights) using a t test (which we ll cove in Pactical ), you can calculate, and theefoe the popotion of the vaiability in body weight explained by sex. And if you ead the esults of a t test in a eseach aticle, you can intepet them in tems of using this technique. Coelations when the dichotomous vaiable is atificial The male/female dichotomy is natual; all subjects ae eithe one o the othe. Sometimes a dichotomy is abitay, such as pass/fail in an exam with a 60% pass mak; this dichotomy classifies people who scoed 59% and people who scoed 1% in the same categoy, but classifies people who scoed 59% and people who scoed 60% in diffeent categoies. If you have data like these and want to calculate a coelation, you have to use a slightly diffeent technique; this is descibed by Howell (1997, p. 86). Coelations with two dichotomous vaiables If you want to calculate the coelation between two vaiables when both ae dichotomous, again, you can do it. All you do is calculate in the nomal way; this time, its special name is φ (phi). And it s exactly equivalent to doing a χ test (which we ll cove in Pactical 4). And thee s a elationship between the two: φ χ n

12 1 Why is this useful? Again, because is a measue of the popotion in the vaiability in one vaiable that s explained by a vaiable the pactical significance of the elationship and theefoe so is φ. So if you see a χ test epoted in an aticle, you can calculate φ to see whethe the elationship is impotant (lage) as well as significant; see Howell (1997, p. 85). Is a egession slope (b) significantly diffeent fom zeo? We saw ealie how to test if a coelation () was significantly diffeent fom 0. Since coelation and egession ae much the same thing, we can also calculate the t statistic fom the egession paamete b (fom y ˆ bx + a ) using a diffeent fonula. Fo this it helps to use the notation s X athe than sesidual fo the standad deviation of the esiduals left ove when we have pedicted fom X. We would find that the t statistic we ve just woked out could also be found like this: t n b sx s X n 1 As befoe, this t statistic is distibuted with n degees of feedom (which is why the subscipt on the t is n ). If we calculate two egessions, ae they significantly diffeent? Suppose we calculate the elationship between smoking and life expectancy in males and females. We d pobably find that the moe you smoke, the shote you live (b < 0). Let s suppose we find that this elationship is stonge in males (e.g. b male < b female < 0), suggesting that males decease thei life expectancy moe than females fo a given incement in the amount they smoke (though, of couse, the egession by itself doesn t tell you anything about causality). Is this diffeence between males and females significant? If we have two vaiables X 1 and X that both pedict a thid vaiable, and two sample egession coefficients b 1 and b, then we can calculate a t statistic (with n 4 df) fo the null hypothesis that the two undelying population egession coefficients ae the same: b1 b tn 4 s X s 1 X + s n 1) s ( n 1) X ( 1 X 1 I have two values of fom diffeent (independent) goups. Ae they diffeent? If we want to do the same with coelations athe than egessions ( is the coelation 1 between male smoking and male life expectancy significantly diffeent fom the coelation between female smoking and female life expectancy? ) we have to use a slightly diffeent test. We convet to a elated numbe and wok out a Z scoe fom those: ln z n 3 n Then we look up ou value of z in a table of the standad nomal distibution to get ou p value. I have two values of, but they ae not independent; ae they diffeent? Suppose we measued the numbe of GCSE points acquied by a goup of 16-yeaolds, then measue the numbe of A-Level points acquied by the same people aged 18, then measue thei annual income when they ae 30. We could calculate a coelation between any two of these vaiables. We could also ask whethe the coelation between GCSE scoes and income was bette/wose than the coelation between A-

13 13 Level scoes and income. But these ae clealy not independent coelations, because they wee all based on the same people, and so thee will pobably be a coelation between GCSE scoes and A-Level scoes that we must take into account. If ou thee coelations ae A, B, and C, and we want to know if the diffeence between A and B is significant, then the null hypothesis is that A and B ae the same, and we can calculate a t statistic (with n 3 df) like this: AB AC BC R (1 ) + AB AC BC t n 3 ( AB AC ) n 1 ( R + n 3 ( n 1)(1 + AB + 4 ) BC AC ) (1 BC ) 3 This is effectively a statistical test fo patial coelations. The patial coelation coefficient will answe the question what is the coelation between X and, taking account of Z? This test will answe the question is the coelation xy significantly diffeent fom the coelation xz, taking into account the fact that these two coelations ae themselves elated (non-independent)? Is my value of diffeent fom (a paticula value)? Suppose we have a sample with a coelation of 0.3, and we want to know if this diffes fom a coelation of 0.5. The null hypothesis is that the sample 0.3 came fom a population with ρ 0.5. We can calculate a Z scoe like this: ln ρ z 1 n 3 If I calculate, what ae the confidence limits on ρ? The expession fo z above tells us that z ρ + n 3 so we can calculate confidence intevals fom appopiate citical values of z fo a two-tailed α: 1 CI ( ρ ) ± zα / n 3 If you want 95% confidence intevals, z α/ would be Once you ve woked out confidence intevals fo ρ, we can convet them back to ρ to get ou final answe: ρ 1+ ρ e 1 ρ 0.5ln ρ ρ ρ e + 1 I have a goup of subjects and have woked out a coelation fo each one. Is this coelation significant fo my whole goup? Stop and go back a stage. Suppose you have a goup of 0 ats and you measue thei pefomance on a test of attention and, simultaneously, the levels of the neuotansmitte acetylcholine in pats of thei bain. Is thee a elationship between acetylcholine and attentional pefomance? If you only make one measuement pe at, the poblem is easy; you have 0 measuements of two vaiables, and can coelate them as usual. If you ve made 100 measuements fo each at, the poblem is hade. What can you do? ou must not lump all the measuements togethe to give 000 diffeent {x, y} pais because these obsevations ae definitely not equally independent, since subsets of obsevations ae likely to be elated by vitue of having come fom the same at.

14 14 To ask whethe subjects with high levels of acetylcholine have high levels of pefomance a between-subjects question you could take each at s mean pefomance and mean acetylcholine and conduct a coelation as nomal (n 0). (And if you d made 60 obsevations on some ats and 105 obsevations on othes, it wouldn t matte, because you d take the mean acoss all these subjects. If you eally felt that it was woth placing moe weight on data fom subjects that you obtained measuements fom, you could conduct a weighted analysis, weighting fo the numbe of obsevations pe subject.) If diffeent ats have vey diffeent levels of acetylcholine, then we could end up with something like ou wild-boa-and-unne-bean effect fo example, you might find a negative coelation acoss the goup (ats with lots of acetylcholine do wose than ats with less acetylcholine), even though if you looked fo it, you might find a positive coelation within each at (when any given at has what is a high level of acetylcholine fo that at, it pefoms bette). So If we want to know whethe changes in one vaiable (acetylcholine) ae paalleled by changes in the othe vaiable (pefomance) in the same subject, and that this is consistent acoss subjects a within-subjects question using data fom multiple subjects we can estimate the elationship within subjects using a vey geneal technique, called geneal linea modelling. This paticula way of using a geneal linea model (GLM) is called multiple egession o analysis of covaiance (ANCOVA). The GLM technique will handle even moe complicated poblems, such as when we have two goups of ats (a contol goup and one that has had pat of thei bain destoyed) and we want to know whethe the elationship between acetylcholine and pefomance is diffeent in the two goups. We will not cove these advanced techniques in the IB couse. I want to pedict a vaiable on the basis of many othe vaiables, not just one. Then you need multiple egession, which we e not going to cove. Coelation/egession in Excel elevant functions (see Excel help fo full details) COVAR( ) Population covaiance (i.e. divide-by-n fomula). So to calculate using this numbe, you need to divide this by the poduct of the population SDs of X and, calculated using STDEVP( ) o multiply COVAR( ) by n and then divide it by n 1, befoe dividing the esult by the poduct of the sample SDs, calculated using STDEV( ). CORREL( ) Calculates. RANK( ) Don t use it it gets the anks wong when thee ae ties. Tools Data Analysis Covaiance Calculates sample covaiances (i.e. divide-by-n 1 fomula). Tools Data Analysis Coelation Calculates. Tools Data Analysis Regession Calculates, a, b, p. Bibliogaphy Howell, D. C. (1997). Statistical Methods fo Psychology. Fouth edition, Wadswoth, Belmont, Califonia.