EXST7015 : Statistical Techniques II ANOVA Design Identification Page 1

Transcription

1 NOV Desgn Identfcaton Page 1 Expermental Desgn Identfcaton To correctly desgn an experment, or to analyze a desgned experment, you must be able to look at a desgn stuaton and correctly assess the salent aspects of the desgn. I wll ask you to dentfy the desgn, the treatment varable, dependent varable, degrees of freedom error, expermental unt, samplng unt (f any), and f the treatment s fxed or random. To begn wth, determne what the nvestgators are tryng to do and what they plan to measure. What s the Objectve of the study? pecfcally, what hypotheses are to be tested? What s the varable of nterest? What unt s the treatment appled to? What, exactly, s the unt measured? uppose an nvestgator wants to compare the oxygen levels n seven predefned "habtats" n the Lousana marsh. He wll randomly select and sample 4 stes n each habtat. One oxygen measurement s made at each ste. What varable s beng measured? Ths varable wll produce a seres of measurements or quanttes? Oxygen levels (usually n ppm) What are the treatments? What s the nvestgator nterested n comparng or testng for dfferences? Habtats (t=7) re there any blocks (.e. sources of varaton that should be recognzed, but whch are not mportant to the nvestgator). For example, dd he replcate the experment n several dfferent rvers or several locatons along the coast? re the 4 replcates just multple observatons or are they taken n 4 separate places? pparently no blocks. What are the expermental unts for the experment? What unt was the treatment appled to or what was sampled for each treatment (habtat)? ste (s=4) re there separate samplng unts at each ste, or s only one measurement taken n each expermental unt? In ths case t s a water sample on whch oxygen s measured. nce there s only a sngle sample at each ste we can consder each sample to represent the ste. lso the stes If there were multple samples taken at each ste these would be the samplng unts. These n turn can be splt nto sub-samplng unts. pparently ths was not done. Is that all? There are other ssues, not all of whch we have covered. re the treatments fxed or random? Is the desgn balanced? re there any partcular hypothess tests of nterest (contrasts)? re the treatment levels quanttatve? ny other specal post NOV applcatons? The topc of Desgn wll be dscussed n the second half of the course. For the moment our objectve s only to learn to dentfy the components of an experment. Therefore, I wll put a desgn descrpton on the Internet and durng each class perod I expect you to have looked at t and to be prepared to answer the followng questons.

2 NOV Desgn Identfcaton Page Questons: 1) What s the treatment arrangement for ths experment? (a) sngle factor (b) factoral (c) nested ) What s the expermental desgn for ths experment? (a) RD (b) RD (c) LD (e) plt-plot (d) Repeated Measures 3) Does t seem more lkely that the treatments are fxed or random? (a) fxed (b) random 4) What s the expermental unt for ths experment? (a) pens (b) dets (c) lve weght (d) egg yolk weght (e) ndvdual chckens 5) What s the samplng unt for ths experment? (a) pens (b) dets (c) lve weght (d) egg yolk weght (e) ndvdual chckens 6) What s the dependent varable for ths experment? (a) pens (b) dets (c) lve weght (d) egg yolk weght (e) ndvdual chckens 7) What s the treatment varable for ths experment? (a) pens (b) dets (c) lve weght (d) egg yolk weght (e) ndvdual chckens 8) If the desgn s RD, what are the blocks? (a) pens (b) dets (c) lve weght (d) egg yolk weght (e) ndvdual chckens (f) N 9 & 10) How many degrees of freedom are avalable for testng the treatment (combnatons)? Enter the correct value here: numerator =, denomnator = For the lttle experment dscussed the source table s: ource d.f. Treatment 6 Error 1 Total 7 fnal note on the Daly Desgns. These wll start early n the semester wth the smpler desgns and progress to more complcated desgns. Our only objectve wth these desgns s that you learn to dentfy the mportant aspects and components of the desgns. Durng the second half of the course we wll dscuss the desgns n detal. We wll see why these components exst, how they are analyzed and how they are nterpreted. t the start of each class I wll roll a dce. If a value of 6 s obtaned we wll have a quz. If any other value s rolled I wll smply gve you the answers to the quz. If a value other than 6 s rolled, then that number s not to be counted agan untl after a 6 has occurred. For example, f I roll a 3 then any values of 3 on subsequent days would not count untl a 6 occurred. Once a 6 s rolled all numbers are back n contenton.

3 NOV Desgn Identfcaton Page 3 Identfyng desgns The smplest type of desgn s the ompletely Randomzed Desgn (RD) Ths wll consst of: 1) a treatment (at least one, possbly more), and ) an error term (at least one, possbly more) Example 1: researcher s studyng the sze of tomatoes from plants grown under three waterng regmes, () daly waterng, () waterng at day ntervals and () waterng at 3 day ntervals. Eghteen plants are planted n large pots and 6 are randomly selected for each waterng regme. Plant productvty s measured as the total weght of the frst 10 tomatoes (n cm) produced by each plant or pot (these are synonymous for ths study snce there s only one plant per pot). Dagnoss: What are the treatments and expermental unts? The objectve s to compare the total weght of tomatoes. The varable of nterest (dependent varable) s the total weght of the tomatoes. The treatment s the varable whose levels are to be compared. In ths case the treatment has 3 levels, (.e. the 3 waterng regmes). The expermental unt s the entty that s assgned a treatment. In ths case the treatments are randomly assgned to pots / plants (synonymous n ths case). Note that the weght values we analyze are NOT for ndvdual tomatoes, but are the mean for each plant. nce we are nterested n the mean for each plant we wll have a dataset wth only one value for each plant or pot, so we consder pot/plant to be the samplng unt. In ths partcular experment the pots or plants are both the samplng unt and the expermental unt. In other experments the tomatoes could be measured ndvdually and the samplng unt would be ndvdual tomatoes. However, when the ndvdual unts are summed or averaged to gve just one measurement for each plant, the plant would be consdered the samplng unt. nce the desgn does not have ndvdual tomatoes we have the followng. Descrpton: The model s Y j = + + j Ths analyss s a RD wth a sngle factor treatment and an expermental error. Each treatment was replcated 6 tmes. The source table: ource d.f. EM Waterng Regme t 1 = σ + Q τ Plant (Regme) t(n 1) = 15 σ Total tn 1 = 17

4 NOV Desgn Identfcaton Page 4 Random versus Fxed treatment effects: There are a few other aspects of a desgn that wll be mportant to dentfyng desgns and achevng the correct analyss. One of these s to determne f treatment levels are randomly chosen from a large (theoretcally nfnte) number of choces, or f the treatment levels represent all possble levels, or at least all levels of nterest. Treatment levels that are randomly chosen from a large number of possble levels estmate the varablty among the levels, and are actually estmates of varance components. These are represented as, and represent the varablty among all the levels of the treatment. When the levels of the treatment represent all possble levels, or f they are chosen by the nvestgator as the only levels of nterest they are called fxed effects, and statstcal nference s lmted to the treatment levels ncluded n the experment. Fxed effect treatments do not estmate varances, but rather the sum of squared treatment effects (devatons of each treatment level from the overall mean,.e. n These are summed ( τ Y Y. 1 ) and are represented as Q τ τ Y Y ).. There are obvously many possble waterng regmes, but the 3 of nterest above do not appear to be randomly chosen. They would be fxed. For most of the smple analyses that we wll examne the analytcal dfferences between fxed effects and random effects wll not be obvous. However, for larger expermental desgns the dfferences become very mportant, so we wll contnue to pay attenton to ths aspect of each analyss. Example : uppose we modfed the experment above slghtly by mantanng ndvdual measurements for each tomato. Now, nstead of 18 measurements (one mean per plant) we would have 180 measurements, one weght for each of 10 tomatoes on each of the 18 plants. How would ths change the desgn? Dagnoss: What are the treatments and expermental unts? The objectve s to compare the weght of tomatoes, but n ths case the varable of nterest (dependent varable) s the weght of ndvdual tomatoes. The treatment s stll the 3 waterng regmes. The expermental unt (the entty that s assgned a treatment) s stll the pots or plants. The samplng unt s now the ndvdual tomatoes, snce we now measure each tomato ndvdually and record a separate measurement for each tomato frut.

5 NOV Desgn Identfcaton Page 5 Descrpton: The model s Y jk = + + j + jk Ths analyss s a RD wth a sngle factor treatment and wth both an expermental error for the expermental unts and a samplng error for the samplng unts. Each treatment was replcated 6 tmes and has 10 samplng unts per expermental unt. The source table: ource d.f. EM Regme t 1 = σ + nσ γ + Q τ Plant (Regme) t(p 1) = 15 σ + nσ γ Tomato (Plant x Regme) tp(n 1) = 16 σ Total tpn 1 = 179 Note that the σ estmates the component of varablty among tomatoes and σ γ estmates the component of varablty among the pots or plants. These two sources of varablty (among tomatoes and among pots) may or may not dffer (.e. σ γ may equal zero). lso note that the approprate error term for the waterng regme s the expermental error term. Example 3: uppose the room where we were to culture the plants has hghly varable condtons. In partcular there s a strong east to west varaton n temperature and lght condtons. If we place the treatments completely at random, we may get too many of one treatment on the east, where condtons are better, makng that treatment look better than t really s. lso, the dfferng condtons wll cause extra varablty among the plants producton and, therefore, the tomato weghts. Ths extra varablty wll be added to the error term, reducng the power of the experment (tests are the most powerful when the error term s small and the treatment dfferences are large). If we gnore the east-west varaton t wll become part of the error term. o we decde to place our treatments n 6 groups. Each group contans only 1 replcate of the each treatment, but the 6 groups themselves consttute our replcaton. Note that the 6 groups could be any type of groupng that accounts for varaton. For example, f we only had 3 sutable pots or chambers for the experment we could replcate the experment n 6 tme perods. s before, n the frst experment, we wll take the total weght of 10 tomatoes per plant for comparng the plants. Group 1 Group Group 3 Group 4 Group 5 Group 6

6 NOV Desgn Identfcaton Page 6 Dagnoss: What are the treatments and expermental unts? The objectve s to compare tomato total weghts. The varable of nterest s the total weght of 10 tomatoes, as before. The treatment s stll the 3 waterng regmes. The expermental unt s stll the pots or plants, and these also the samplng unts whch agan provde one measurement for each plant. The new wrnkle s the 6 dfferent groups on whch we conduct our experment. Due to the potental dfferences (east-west), we wll want to remove ths varaton from the error term. We do ths by blockng on group and actually puttng a separate varable n the model to account for block dfferences. If we don t, ths addtonal varaton among groups (east-west) goes nto the error term and reduces our power. Descrpton: The model s Y j = + + j + j Ths analyss s a Randomzed lock Desgn (RD) wth a sngle factor treatment and an expermental error. The source table: ource d.f. EM lock b 1 = 5 σ + tσ β Regme t 1 = σ + Q τ lock x Regme (t 1)(b 1) = 10 σ Total tb 1 = 17 Example 4: uppose we now combne the second and thrd desgn. We conduct the experment wth only 3 pots n each of the 6 groups, but we measure and record 10 ndvdual tomato weghts from each plant. Dagnoss: What are the treatments and expermental unts? The objectve s to compare tomato weghts, and we wll agan do ths wth ndvdual tomatoes. The treatment s stll the 3 waterng regmes. The expermental unt s stll the pots or plants, and the samplng unts (ndvdual tomatoes) agan provde one measurement for each plant. gan we are blockng on the groupngs. Descrpton: The model s Y jk = + + j + j + jk Ths analyss s a Randomzed lock Desgn (RD) wth a sngle factor treatment and wth both an expermental error and a samplng error. The source table: ource d.f. EM lock b 1 = 5 σ + nσ βτ + ntσ β Regme t 1 = σ + nσ βτ + Q τ lock x Regme (t 1)(b 1) = 10 σ + nσ Tomato (lock x Regme) tb(n 1) = 16 Total tbn 1 = 179

7 NOV Desgn Identfcaton Page 7 Example 5: One last example. uppose we go back to our frst experment, the RD where we take the total weght of 10 tomatoes per plant. However, suppose we have two treatments of nterest nstead of the sngle factor (regme). The two treatments of nterest are (a) the 3 waterng regmes and we are also nterested n comparng between plants that receve a low level of fertlzer and those that receve a hgh level. We wll examne all 6 combnatons of treatments, and 3 of the 18 potted plants wll be allocated to each treatment combnaton. Notce that the treatments are cross classfed. Each level of Waterng occurs wth each level of Fertlzer such that all possble combnatons exst. Ths s characterstc of a factoral analyss of varance, also called a two-way NOV. Fertlzer treatment ( levels) Waterng regme treatment (3 levels) daly waterng () -day ntervals () 3-day ntervals () Low (1) 3 replcate pots 3 replcate pots 3 replcate pots Hgh () 3 replcate pots 3 replcate pots 3 replcate pots Dagnoss: What are the treatments and expermental unts? The objectve s to compare tomato weghts, and we wll have one mean per plant. There are now two treatments, one wth levels and one wth 3 levels. Ths s called a x3 factoral treatment arrangement. Note that low fertlzer and hgh fertlzer would appear to cover all possble levels of ths treatment, so ths treatment s also fxed. oth the expermental unts and samplng unts are stll the pots / plants. There s no blockng n ths experment, so ths analyss s a RD wth a x3 factoral treatment arrangement. Descrpton: The model s Y jk = + + j + j + jk The source table: ource d.f. EM Fertlzer t 1 1 = 1 σ + Q Regme t 1 = σ + Q Fertlzer x Regme (t 1 1)(t 1) = σ + Q Pot (Fertlzer x Regme) t 1 t (n 1) = 1 σ Total t 1 t n 1 = 17 τ1 τ τ1τ

8 NOV Desgn Identfcaton Page 8 lternatvely: let treatments represent all t=6 combnatons of t 1 by t ource d.f. Treatments t 1 = 5 Expermental Error t(n 1) = 1 Total tn 1 = 17 ome fnal notes on the dentfcaton of desgned experments. There are other types of desgns that we wll dscuss ths semester. We wll dscuss the Latn quare Desgn (LD), whch has a pecular structure wth two sources of blockng (genercally referred to as rows and columns ). The treatments are arranged n such a way that each treatment occurs once n each row and once n each column. nother major class of desgns wll be dscussng under the ttle of plt plots and Repeated measures. Ths class of desgns has an ntal structure that can be RD, RD or LD wth treatment arrangements that may be ether a sngle factor or factoral. Then each expermental unt s ether splt nto two or more unts and a new treatment appled to the sub-unts of the expermental unt, or each expermental unt s sampled over tme. In the latter case tme becomes a source of varaton of nterest, and s ncluded n the source table. For example, n our tomato plant example above we mght splt our expermental unt (the plant) nto two levels, the lower half of the plant and the upper half of the plant. We mght take 10 tomatoes from the lower half and 10 tomatoes from the upper half. The new varable level would be ncluded n the experment. Ths would be an example of a splt plot. nother possblty s that we are nterested n the changng sze of the tomatoes produced by each plant over tme. We mght take the frst 10 tomatoes (tme 1), and the second 10 tomatoes (tme ), etc. Ths would gve a new varable called tme that would show f the sze of tomatoes was the same over tme or not, and f the dfferent waterng regmes were smlar of dfferent over tme. There s also another treatment arrangement called the nested treatment arrangement. Ths arrangement has a herarchcal arrangement of treatments wth one level nested wthn the hgher level. Do not confuse ths treatment arrangement wth the nested error terms dscussed above where the samplng error s also nested wthn the expermental error n a herarchcal fashon. lso note that t s possble to have many levels of nestng of both treatments and error terms. Random effects versus fxed effects. When the effects are fxed we are usually nterested n the ndvdual treatment levels. Frequently, we wll calculate a mean for each treatment level and do comparsons and tests among those means. Random effects, however, represent a random selecton from a very large number of possble levels, and we are not usually nterested n each of the ndvdual levels selected. For random effects we are most lkely nterested n the overall mean and the varablty about that mean, so we would lkely want to place a confdence nterval on that mean.

9 NOV Desgn Identfcaton Page 9 Introducton Major topcs (a comprehensve outlne s provded elsewhere) Regresson : LR, Multple, urvlnear & Logstc Expermental Desgn : RD, RD, LD, plt-plot & Repeated Measures Treatment arrangements : ngle factor, Factoral, Nested ourse Objectves The objectves of the ntroductory course were to develop an understandng of elemental statstcs, the ablty to understand and apply basc statstcal procedures. We wll develop those concepts further, applyng the termnology and notaton from the basc methods courses to advanced technques for makng statstcal nferences. We wll cover the major methodologes of parametrc statstcs used for predcton and hypotheses testng (prmarly regresson and expermental desgn). Our emphass wll be on REOGNIZING analytcal problems and on beng able to do the statstcal analyss wth software. We wll see programs and output for vrtually all analyses covered ths semester. Daly Desgn I wll be placng a desgn descrpton on the Internet for each class. You should plan on examnng ths desgn before class. t the begnnng of each class I wll randomly determne whether we have a quz on that desgn or not. I do not ntend to spend much tme on ths daly actvty. If there s a quz, I wll allow 5 mnutes for you to answer and turn n quz. If not, I wll gve you the answers. quz wll consst of specfyng the dependent varable, expermental and samplng unts, treatments, blocks, random effects, etc. We wll address most of these n the desgn secton of the course (followng regresson). However, some wll be covered n the daly desgn. Notes on Exams I usually schedule a revew sesson late on Tuesday for the Thursday exams and unday for Tuesday exams. Revew sesson s entrely voluntary, and you may leave anytme. I wll have not plan on coverng materal. I plan only to answer questons. There wll be no revew for the fnal exam. On the exam you wll be allowed to brng a calculator I do not expect to have many calculatons on the exam, but there may be some. For example, calculatng a t-test for a slope for an hypotheszed value other than zero (thought ths may be n the output, always check frst). lso confdence ntervals on slopes and treatment means. You may also brng an 8.5 by 11 nch sheet of paper wth equatons or whatever else you wsh to nclude. You may wrte on both sdes of that pece of paper. I wll provde you wth these on an exam. You wll need to understand MY t-tables. ee nterment for copes of these tables. ll exams, ncludng the fnal, wll be n our regular classroom(s).

10 NOV Desgn Identfcaton Page 10 mple Lnear Regresson (revew?) The objectve: Gven ponts plotted on two coordnates, Y and X, fnd the best lne to ft the data. Y - the dependent varable X - the ndependent varable The concept: Data conssts of pared observatons wth a presumed potental for the exstence of some underlyng relatonshp. We wsh to determne the nature of the relatonshp and quantfy t f t exsts. Note that we cannot prove that the relatonshp exsts by usng regresson (.e. we cannot prove cause and effect). Regresson can only show f a correlaton exsts, and provde an equaton for the relatonshp. Gven a data set consstng of pared, quanttatve varables, and recognzng that there s varaton n the data set, we wll defne, POPULTION MODEL (LR): Y 0 1X Ths s the model we wll ft. It s the equaton descrbng straght lne for a populaton and we want to estmate the parameters n the equaton. The populaton parameters to be estmated are for the underlyng model, yx. 0 1X, are: Termnology yx. = the true populaton mean of Y at each value of X 0 = the true value of the Y ntercept 1 = the true value of the slope, the change n Y per unt of X Dependent varable: varable to be predcted Y = dependent varable (all varaton occurs n Y) Independent varable: predctor or regressor varable X = ndependent varable (X s measured wthout error) Intercept: value of Y when X = 0, pont where the regresson lne passes through the Y axs. The unts on the ntercept are the same as the Y unts lope: the value of the change n Y for each unt ncrease n X. The unts on the slope are Y unts per X unt Devaton: dstance from an observed pont to the regresson lne, also called a resdual.

11 NOV Desgn Identfcaton Page 11 Least squares regresson lne: the lne that mnmzes the squared dstances from the lne to the ndvdual observatons. Regresson lne Y - the dependent varable Intercept Devatons X - the ndependent varable The regresson lne tself represents the mean of Y at each value of X ( yx. ). Regresson calculatons ll calculatons for smple lnear regresson start wth the same values. These are, n n n n n,,,,, X X Y Y X Y n alculatons for smple lnear regresson are frst adjusted for the mean. These are called corrected values. They are corrected for the MEN by subtractng a correcton factor. s a result, all smple lnear regressons are adjusted for the mean of X and Y and pass through the pont Y, X. Y Y, X X The orgnal sums and sums of squares of Y are dstances and squared dstances from zero. These are referred to as uncorrected meanng unadjusted for the mean. Y 0 X

12 NOV Desgn Identfcaton Page 1 The corrected devatons sum to zero (half negatve and half postve) and the sums of the squares are squared dstances from the mean of Y. Y Y X Once the means, obtaned, the calculatons for the parameter estmates are: lope = b XY 1 X Y and corrected sums of squares and cross products,, Intercept = b0 Y b1x We have ftted the sample equaton are YY XY Y b0 b1x e, whch estmates the populaton parameters of the model, Y 0 1X Varance estmates for regresson fter the regresson lne s ftted, varance calculatons are based on the devatons from the regresson. From the regresson model Y b0 b1x e we derve the formula for the devatons e Y b b X or e = Y-Y ˆ. 0 1 Y X s wth other calculatons of varance, we calculate a sum of squares (corrected for the mean). Ths s smplfed by the fact that the devatons, or resduals, already have a mean of zero, n e 1 Resduals = = Error. The degrees of freedom (d.f.) for the varance calculaton s n, snce two parameters are estmated pror to the varance ( 0 and 1 ). The varance estmate s called the ME (Mean square error). It s the Error dvded by the d.f., ME E n. The varances for the two parameter estmates and the predcted values are all dfferent, but all are based on the ME, and all have n d.f. (t-tests) or n d.f. for the denomnator (F tests). Varance of the slope = ME

13 NOV Desgn Identfcaton Page 13 Varance of the ntercept = 1 X ME n Varance of a predcted value at X = ME n X X 1 ny of these varances can be used for a t-test of an estmate aganst an hypotheszed value for the approprate parameter (.e. slope, ntercept or predcted value respectvely). NOV table for regresson common representaton of regresson results s an NOV table. Gven the Error (sum of squared devatons from the regresson), and the ntal total sum of squares ( YY ), the sum of squares of Y adjusted for the mean, we can construct an NOV table mple Lnear Regresson NOV table d.f. um of quares Mean quare F Regresson 1 Regresson MReg Error n Error MError Total n 1 YY = Total MReg / MError In the NOV table The Regresson and Error sum to the Total, so gven the total ( YY ) and one of the two terms, we can get the other. The easest to calculate frst s usually the Regresson snce we usually already have the necessary ntermedate values. Regresson = XY The Regresson s a measure of the mprovement n the ft due to the regresson lne. The devatons start at YY and are reduced to Error. The dfference s the mprovement, and s equal to the Regresson. Ths gves another statstc called the R. What porton of the Total ( YY ) s accounted for by the regresson? R = Regresson / Total The degrees of freedom n the NOV table are, n 1 for the total, one lost for the correcton for the mean (whch also fts the ntercept) n for the error, snce two parameters are estmated to get the regresson lne. 1 d.f. for the regresson, whch s the d.f. for the slope. tatstcs quote: He uses statstcs as a drunken man uses lampposts -- for support rather than for llumnaton. ndrew Lang ( ), cottsh poet, folklorst, bographer, translator, novelst, and scholar

14 NOV Desgn Identfcaton Page 14 The F test s constructed by calculatng the MRegresson / MError. Ths F test has 1 n the numerator and (n ) d.f. n the denomnator. Ths s exactly the same test as the t-test of the slope aganst zero. To test the slope aganst an hypotheszed value (say zero) usng the t-test wth n d.f., calculate b1 b1 Hypotheszed b1 0 t b ME 1 ssumptons for the Regresson We wll recognze 4 assumptons 1) Normalty We take the devatons from regresson and pool them all together nto one estmate of varance. ome of the tests we use requre the assumpton of normalty, so these devatons should be normally dstrbuted. Y X For each value of X there s a populaton of values for the varable Y (normally dstrbuted). ) Homogenety of varance When we pool these devatons (varances) we also assume that the varances are the same at each value of X. In some cases ths s not true, partcularly when the varance ncreases as X ncreases. 3) X s measured wthout error! nce varances are measured only vertcally, all varance s n Y, no provsons are made for varance n X. 4) Independence. Ths enters n several places. Frst, the observatons should be ndependent of each other (.e. the value of e should be ndependent of e j, for j). lso, n the equaton for the lne Y b0 b1x e we assume that the term e s ndependent of the rest of the model. We wll talk more of ths when we get to multple regresson. o the four assumptons are: Normalty Homogenety of varance Independence X measured wthout error These are explct assumptons, and we wll examne or test these assumptons when possble. There are also some other assumptons that I consder mplct. We wll not state these, but n some cases they can be tested. For example, There s order n the Unverse. Otherwse, what are you nvestgatng? haos? The underlyng fundamental relatonshp that I just ftted a straght lne to really s a straght lne. ometmes ths one can be examned statstcally.

15 NOV Desgn Identfcaton Page 15 haracterstcs of a Regresson Lne The lne wll pass through the pont Y, X (also the pont b 0, 0) The sum of devatons wll be zero ( e 0 ) 0 1 of the ponts from the regresson lne wll be a mnmum. Values on the lne can be descrbed by the equatonyˆ b0 b1x. The lne has some desrable propertes (f the assumptons are met) E b The sum of squared devatons (measured vertcally, e Y b bx o o E b E Y X Y X o. Therefore, the parameter estmates and predcted values are unbased estmates. Note that lnear regresson s consdered statstcally robust. That s, the tests of hypothess tend to gve good results f the assumptons are not volated to a great extent. rossproducts and correlaton rossproducts are used n a number of related calculatons (can be + or ). a crossproduct = YX um of crossproducts = YX orrected sum of crossproducts = XY ovarance = lope = XY XY Regresson = orrelaton = XY R =r = XY YY n 1 XY mple Lnear Regresson ummary YY = Regresson / Total ee mple lnear regresson notes from EXT7005 for addtonal nformaton, ncludng the dervaton of the equatons for the slope and ntercept. You are not responsble for these dervatons. Know the termnology, characterstcs and propertes of a regresson lne, the assumptons, and the components to the NOV table. You wll not be fttng regressons by hand, but I wll expect you to understand where the values on output come from and what they mean. Partcular emphass wll be placed on workng wth, and nterpretng, numercal regresson analyses. nalyses wll mostly be done wth.