NCSS Statistical Software. Poisson Regression

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1 NCSS Statstcal Software Chapter 325 Posson Regresson Introducton Posson regresson s smlar to regular multple regresson except that the dependent (Y) varable s an observed count that follows the Posson dstrbuton. Thus, the possble values of Y are the nonnegatve ntegers: 0,, 2, 3, and so on. It s assumed that large counts are rare. Hence, Posson regresson s smlar to logstc regresson, whch also has a dscrete response varable. However, the response s not lmted to specfc values as t s n logstc regresson. One example of an approprate applcaton of Posson regresson s a study of how the colony counts of bactera are related to varous envronmental condtons and dlutons. Another example s the number of falures for a certan machne at varous operatng condtons. Stll another example s vtal statstcs concernng nfant mortalty or cancer ncdence among groups wth dfferent demographcs. Most books on regresson analyss brefly dscuss Posson regresson. We are aware of only one book that s completely dedcated to the dscusson of the topc. Ths s the book by Cameron and Trved (998). Most of the methods presented here were obtaned from ther book. Ths program computes Posson regresson on both numerc and categorcal varables. It reports on the regresson equaton as well as the goodness of ft, confdence lmts, lkelhood, and devance. It performs a comprehensve resdual analyss ncludng dagnostc resdual reports and plots. It can perform a subset selecton search, lookng for the best regresson model wth the fewest ndependent varables. It provdes confdence ntervals on predcted values. The Posson Dstrbuton The Posson dstrbuton models the probablty of y events (.e. falure, death, or exstence) wth the formula µ e µ Pr = y! ( Y = y µ ) = ( y 0,,2,... ) Notce that the Posson dstrbuton s specfed wth a sngle parameter µ. Ths s the mean ncdence rate of a rare event per unt of exposure. Exposure may be tme, space, dstance, area, volume, or populaton sze. Because exposure s often a perod of tme, we use the symbol t to represent the exposure. When no exposure value s gven, t s assumed to be one. The parameter µ may be nterpreted as the rsk of a new occurrence of the event durng a specfed exposure perod, t. The probablty of y events s then gven by y ( µ t) e Pr ( Y = y µ, t) = ( y = 0,,2,... ) y! The Posson dstrbuton has the property that ts mean and varance are equal. µ t y 325- NCSS, LLC. All Rghts Reserved.

2 NCSS Statstcal Software Posson Regresson The Posson Regresson Model In Posson regresson, we suppose that the Posson ncdence rate µ s determned by a set of k regressor varables (the X s). The expresson relatng these quanttes s ( β X + β X + β ) µ = t exp + Note that often, X and β s called the ntercept. The regresson coeffcents β, β2,, βk are unknown parameters that are estmated from a set of data. Ther estmates are labeled b, b 2,, bk. Usng ths notaton, the fundamental Posson regresson model for an observaton s wrtten as where Pr µ = t µ 2 ( Y = y µ, t ) = t exp 2 e = µ t ( t ) µ y! k X k ( x β) ( β X + β X + + β X ) That s, for a gven set of values of the regressor varables, the outcome follows the Posson dstrbuton. Soluton by Maxmum Lkelhood Estmaton The regresson coeffcents are estmated usng the method of maxmum lkelhood. The logarthm of the lkelhood functon s ln n [ L( y, β) ] = y ln[ t µ ( x β) ] t ( x β) ln( y!) = 2 2 n µ = Note that some statstcal packages gnore the last term snce t does not nvolve the regresson parameters. Ths wll make ther calculated log-lkelhoods dfferent from ours. The lkelhood equatons may be formed by takng the dervatves wth respect to each regresson coeffcent and settng the result equal to zero. Dong ths leads to a set of nonlnear equatons that admts no closed-form soluton. Thus, an teratve algorthm must be used to fnd the set of regresson coeffcents that maxmum the loglkelhood. Usng the method of teratvely reweghted least squares, a soluton may be found n fve or sx teratons. However, the algorthm requres a complete pass through the data at each teraton, so t s relatvely slow for problems wth a large number of rows. Wth today s computers, ths s becomng less and less of an ssue. Dstrbuton of the MLE s Applyng the usual maxmum lkelhood theory, the asymptotc dstrbuton of the maxmum lkelhood estmates (MLE s) s multvarate normal. That s, where V βˆ ~ N ( β, βv = n ˆ µ β = ˆβ ) xx Remember that n the Posson model the mean and the varance are equal. In practce, the data almost always reject ths restrcton. Usually, the varance s greater than the mean a stuaton called overdsperson. The y k k n = NCSS, LLC. All Rghts Reserved.

3 NCSS Statstcal Software Posson Regresson ncrease n varance s represented n the model by a constant multple of the varance-covarance matrx. That s, we use where φ s estmated usng V ˆ φ = = n ˆ φ µ β = n n k = x x ( ˆ µ ) NCSS provdes the opton of usng φ (ph) n the calculaton of the varances of the regresson coeffcents. y ˆ µ 2 Goodness of Ft Tests Overall performance of the model s measured by two ch-square tests. These are the Pearson statstc and the devance, or G, statstc D P = P P = n ( y µ ) = µ n y y ln = µ 2 ( y µ ) Both of these statstcs are approxmately ch-square dstrbuted wth n - k degrees of freedom. When a test s rejected, there s a sgnfcant lack of ft. When a test s not rejected, there s no evdence of lack of ft. The Pearson statstc s only ch-square dstrbuted when you are analyzng grouped data, so f you are not usng a frequency varable, you should not use the Pearson statstc as a goodness of ft test. The Pearson statstc s often used as a test of overdsperson. Devance The devance s twce the dfference between the maxmum achevable log-lkelhood and the log-lkelhood of the ftted model. In multple regresson under normalty, the devance s the resdual sum of squares. In the case of Posson regresson, the devance s a generalzaton of the sum of squares. The formula for the devance s D ( y, μˆ ) = 2{ LL LL } y μˆ Pseudo R-Squared Measures The R-squared statstc does not extend to Posson regresson models. Varous pseudo R-squared tests have been proposed. These pseudo measures have the property that, when appled to the lnear model, they match the nterpretaton of the lnear model R-squared. In Posson regresson, the most popular pseudo R-squared measure s functon of the log-lkelhoods of three models 2 LL R = LL ft max LL0 LL NCSS, LLC. All Rghts Reserved.

4 NCSS Statstcal Software where LL Posson Regresson n n n = 0 = y ln [ t ˆ µ ] ˆ µ t ln( y! ) where ˆ µ = n = = = t = LL n n n max = y ln! = = = [ t y ] t y ln( y ) n y LL ft = n = y ln n µ = Note that LL 0 s the log-lkelhood of the ntercept-only model, [ t ˆ µ ( x β) ] t ˆ( x β) ln( y!) n = LL ft s the log-lkelhood of the current model, and LL max s the maxmum log-lkelhood possble. The maxmum log-lkelhood occurs when the actual responses (the y s) exactly equal the predcted responses (the µ s). Notce that ths value of R-squared vares between zero and one, wth a perfect ft occurrng at one. Also note that t assumes that there s an ntercept n the model. Ths may be an actual explct ntercept or an mplct ntercept (as when you use a complete set of ndcator varables to represent a categorcal varable). Resduals As n any regresson analyss, a complete resdual analyss should be employed. Ths nvolves plottng the resduals aganst varous other quanttes such as the regressor varables (to check for outlers and curvature) and the response varable. Varous resduals may be of nterest. These wll be presented next. Raw Resdual The raw resdual s the dfference between the actual response and the estmated value from the model. Because n the Posson case, the varance s equal to the mean, we expect that the varances of the resduals are unequal. Ths can lead to dffcultes n the nterpretaton of the raw resduals. However, they are stll popular. The formula for the raw resdual s r = µˆ y Pearson Resdual The Pearson resdual corrects for the unequal varance n the resduals by dvdng by the standard devaton. The formula for the Pearson resdual s p = y ˆ µ ˆˆ φµ Devance Resdual The devance resdual s another popular resdual. It s popular because the sum of squares of these resduals s the devance statstc. The formula for the devance resdual s ( ) y d = sgn y ˆ µ 2 y ln ( y ˆ µ ) ˆ µ NCSS, LLC. All Rghts Reserved.

5 NCSS Statstcal Software Posson Regresson Hat Values The Hat matrx s used n resdual dagnostcs to measure the nfluence of each observaton. The hat values, h, are the dagonal entres of the Hat matrx whch s calculated usng where W s a dagonal matrx made up of H = W µˆ. / 2 X / 2 ( X ' WX ) X ' W The hat values should be studed themselves, to understand whch observatons have a large nfluence on the ftted regresson coeffcents. Large hat values are those that are larger than 2k/n. They are also used to further standardze resduals as s shown next. Studentzed Pearson Resdual The formula for the studentzed Pearson resdual s sp p = h Studentzed Devance Resdual The formula for the studentzed devance resdual s sd d = h Subset Selecton Subset selecton refers to the task of fndng a small subset of the avalable regressor varables that does a good job of predctng the dependent varable. Because Posson regresson must be solved teratvely, the task of fndng the best subset can be tme consumng. Hence, technques whch look at all possble combnatons of the regressor varables are not feasble. Instead, algorthms that add or remove a varable at each step must be used. Two such searchng algorthms are avalable n ths module: forward selecton and forward selecton wth swtchng. Before dscussng the detals of these two algorthms, t s mportant to comment on a couple of ssues that can come up. The frst ssue s what to do about the bnary varables that are generated for a categorcal ndependent varable. If such a varable has sx categores, fve bnary varables are generated. You can see that wth two or three categorcal varables, a large number of bnary varables may result, whch greatly ncreases the total number of varables that must be searched. To avod ths problem, the algorthms used here search on model terms rather than on the ndvdual varables. Thus, the whole set of bnary varables assocated wth a gven term are consdered together for ncluson n, or deleton from, the model. Its all or none. Because of the tme consumng nature of the algorthm, ths s the only feasble way to deal wth categorcal varables. If you want the subset algorthm to deal wth them ndvdually, you can generate the set of bnary varables manually and desgnate them as Numerc Varables. Herarchcal Models A second ssue s what to do wth nteractons. Usually, an nteracton s not entered n the model unless the ndvdual terms that make up that nteracton are also n the model. For example, the nteracton term A*B*C s not ncluded unless the terms A, B, C, A*B, A*C, and B*C are already n the model. Such models are sad to be herarchcal. You have the opton durng the search to force the algorthm to only consder herarchcal models durng ts search. Thus, f C s not n the model, nteractons nvolvng C are not even consdered. Even though the opton for non-herarchcal models s avalable, we recommend that you only consder herarchcal models NCSS, LLC. All Rghts Reserved.

6 NCSS Statstcal Software Posson Regresson Forward Selecton The method of forward selecton proceeds as follows.. Begn wth no terms n the model. 2. Fnd the term that, when added to the model, acheves the largest value of R-squared. Enter ths term nto the model. 3. Contnue addng terms untl a preset lmt on the maxmum number of terms n the model s reached. Ths method s comparatvely fast, but t does not guarantee that the best model s found except for the frst step when t fnds the best sngle term. You mght use t when you have a large number of observatons so that other, more tme consumng methods, are not feasble, or when you have far too many possble regressor varables and you want to reduce the number of terms n the selecton pool. Forward Selecton wth Swtchng Ths method s smlar to the method of Forward Selecton dscussed above. However, at each step when a term s added, all terms n the model are swtched one at a tme wth all canddate terms not n the model to determne f they ncrease the value of R-squared. If a swtch can be found, t s made and the canddate terms are agan searched to determne f another swtch can be made. When the search for possble swtches does not yeld a canddate, the subset sze s ncreased by one and a new search s begun. The algorthm s termnated when a target subset sze s reached or all terms are ncluded n the model. Dscusson These algorthms usually requre two runs. In the frst run, you set the maxmum subset sze to a large value such as 0. By studyng the Subset Selecton reports from ths run, you can quckly determne the optmum number of terms. You reset the maxmum subset sze to ths number and make the second run. Ths two-step procedure works better than relyng on some F-to-enter and F-to-remove tests whose propertes are not well understood to begn wth. Data Structure At a mnmum, datasets to be analyzed by Posson regresson must contan a dependent varable and one or more ndependent varables. For each categorcal varable, the program generates a set of bnary (0 and ) varables that express the same nformaton. For example, n the table below, the dscrete varable AgeGroup wll be replaced by the varables Ag2 through Ag6 (Ag s not needed). Koch et. al. (986) present the followng data taken from the Thrd Natonal Cancer Survey. Ths dataset contans the number of new melanoma cases n among whte males n two areas for varous age groups. The sze of the estmated populaton at rsk s gven n the varable Populaton NCSS, LLC. All Rghts Reserved.

7 NCSS Statstcal Software Koch36 dataset Posson Regresson Melanoma Area AgeGroup Populaton AG AG2 AG3 AG4 AG5 AG6 6 0 < > < > Mssng Values If mssng values are found n any of the ndependent varables beng used, the row s omtted. If only the value of the dependent varable s mssng, that row wll not be used durng the estmaton process, but ts predcted value wll be generated and reported on. Procedure Optons Ths secton descrbes the optons avalable n ths procedure. Varables, Model Tab Ths panel specfes the varables and model are used n the analyss. Varables Dependent Y Specfy the dependent (response) varable. Ths s the varable to be predcted by the ndependent varables. The values n ths varable should be non-negatve ntegers (zero s okay). Exposure T Specfy an optonal varable contanng exposure values. If ths opton s left blank, all exposures wll be set to.0. Ths varable s specfed when the exposures are dfferent for each row. The exposure s the amount of tme, space, dstance, volume, or populaton sze from whch the dependent varable s counted. For example, exposure may be the tme n days, months, or years durng whch the values on that row were obtaned. It may be the number of ndvduals at rsk or the number of man-years from whch the dependent varable s measured. Each exposure must be a postve (non-zero) number or the row s gnored durng the estmaton phase. Numerc X s Specfy the numerc (contnuous) ndependent varables. By numerc, we mean that the values are numerc and at least ordnal. Nomnal varables, even when coded wth numbers, should be specfed as Categorcal Independent Varables. Although you may specfy bnary (0-) varables here, they are better analyzed when you specfy them as Categorcal Independent Varables NCSS, LLC. All Rghts Reserved.

8 NCSS Statstcal Software Posson Regresson If you want to create powers and cross-products of these varables, specfy an approprate model n the Custom Model feld under the Model tab. If you want to create predcted values of Y for values of X not n your database, add the X values to the bottom of the database. They wll not be used durng estmaton, but predcted values wll be generated for them. Categorcal X s Specfy categorcal (nomnal or group) ndependent varables n ths box. By categorcal we mean that the varable has only a few unque, numerc or text, values lke, 2, 3 or Yes, No, Maybe. The values are used to dentfy categores. Regresson analyss s only defned for numerc varables. Snce categorcal varables are nomnal, they cannot be used drectly n regresson. Instead, an nternal set of numerc varables must be substtuted for each categorcal varable. Suppose a categorcal varable has G categores. NCSS automatcally generates the G- nternal, numerc varables for the analyss. The way these nternal varables are created s determned by the Recodng Scheme and, f needed, the Reference Value. These optons can be entered separately wth each categorcal varable, or they can specfed usng a default value (see Default Recodng Scheme and Default Reference Value below). The syntax for specfyng a categorcal varable s VarName(CType; RefValue) where VarName s the name of the varable, CType s the recodng scheme, and RefValue s the reference value, f needed. CType The recodng scheme s entered as a letter. Possble choces are B, P, R, N, S, L, F, A,, 2, 3, 4, 5, or E. The meanng of each of these letters s as follows. B for bnary (the group wth the reference value s skpped). Example: Categorcal varable Z wth 4 categores. Category D s the reference value. Z B B2 B3 A 0 0 B 0 0 C 0 0 D P for Polynomal of up to 5th order (you cannot use ths opton wth category varables wth more than 6 categores. Example: Categorcal varable Z wth 4 categores. Z P P2 P R to compare each wth the reference value (the group wth the reference value s skpped). Example: Categorcal varable Z wth 4 categores. Category D s the reference value. Z C C2 C3 A 0 0 B 0 0 C 0 0 D NCSS, LLC. All Rghts Reserved.

9 NCSS Statstcal Software Posson Regresson N to compare each wth the next category. Example: Categorcal varable Z wth 4 categores. Z S S2 S S to compare each wth the average of all subsequent values. Example: Categorcal varable Z wth 4 categores. Z S S2 S L to compare each wth the pror category. Example: Categorcal varable Z wth 4 categores. Z S S2 S F to compare each wth the average of all pror categores. Example: Categorcal varable Z wth 4 categores. Z S S2 S A to compare each wth the average of all categores (the Reference Value s skpped). Example: Categorcal varable Z wth 4 categores. Suppose the reference value s 3. Z S S2 S to compare each wth the frst category after sortng. Example: Categorcal varable Z wth 4 categores. Z C C2 C3 A B 0 0 C 0 0 D NCSS, LLC. All Rghts Reserved.

10 NCSS Statstcal Software Posson Regresson 2 to compare each wth the second category after sortng. Example: Categorcal varable Z wth 4 categores. Z C C2 C3 A 0 0 B C 0 0 D to compare each wth the thrd category after sortng. Example: Categorcal varable Z wth 4 categores. Z C C2 C3 A 0 0 B 0 0 C D to compare each wth the fourth category after sortng. Example: Categorcal varable Z wth 4 categores. Z C C2 C3 A 0 0 B 0 0 C 0 0 D to compare each wth the ffth category after sortng. Example: Categorcal varable Z wth 5 categores. Z C C2 C3 C4 A B C D E E to compare each wth the last category after sortng. Example: Categorcal varable Z wth 4 categores. Z C C2 C3 A 0 0 B 0 0 C 0 0 D RefValue A second, optonal argument s the reference value. The reference value s one of the categores. The other categores are compared to t, so t s usually a baselne or control value. If nether a baselne or control value s evdent, the reference value s the most frequent value. For example, suppose you want to nclude a categorcal ndependent varable, State, whch has four values: Texas, Calforna, Florda, and NewYork. Suppose the recodng scheme s specfed as Compare Each wth Reference Value wth the reference value of Calforna. You would enter State(R;Calforna) NCSS, LLC. All Rghts Reserved.

11 NCSS Statstcal Software Posson Regresson Default Recodng Scheme Select the default type of numerc varable that wll be generated when processng categorcal ndependent varables. The values n a categorcal varable are not used drectly n regresson analyss. Instead, a set of numerc varables s automatcally created and substtuted for them. Ths opton allows you to specfy what type of numerc varable wll be created. The optons are outlned n the sectons below. The contrast type may also be desgnated wthn parentheses after the name of each categorcal ndependent varable, n whch case the default contrast type s gnored. If your model ncludes nteractons of categorcal varables, ths opton should be set to Contrast wth Reference or Compare wth All Subsequent' n order to match GLM results for factor effects. Bnary (the group wth the reference value s skpped). Example: Categorcal varable Z wth 4 categores. Category D s the reference value. Z B B2 B3 A 0 0 B 0 0 C 0 0 D Polynomal of up to 5th order (you cannot use ths opton wth category varables wth more than 6 categores. Example: Categorcal varable Z wth 4 categores. Z P P2 P Compare Each wth Reference Value (the group wth the reference value s skpped). Example: Categorcal varable Z wth 4 categores. Category D s the reference value. Z C C2 C3 A 0 0 B 0 0 C 0 0 D Compare Each wth Next. Example: Categorcal varable Z wth 4 categores. Z S S2 S Compare Each wth All Subsequent. Example: Categorcal varable Z wth 4 categores. Z S S2 S NCSS, LLC. All Rghts Reserved.

12 NCSS Statstcal Software Posson Regresson Compare Each wth Pror Example: Categorcal varable Z wth 4 categores. Z S S2 S Compare Each wth All Pror Example: Categorcal varable Z wth 4 categores. Z S S2 S Compare Each wth Average Example: Categorcal varable Z wth 4 categores. Suppose the reference value s 3. Z S S2 S Compare Each wth Frst Example: Categorcal varable Z wth 4 categores. Z C C2 C3 A B 0 0 C 0 0 D 0 0 Compare Each wth Second Example: Categorcal varable Z wth 4 categores. Z C C2 C3 A 0 0 B C 0 0 D 0 0 Compare Each wth Thrd Example: Categorcal varable Z wth 4 categores. Z C C2 C3 A 0 0 B 0 0 C D NCSS, LLC. All Rghts Reserved.

13 NCSS Statstcal Software Posson Regresson Compare Each wth Fourth Example: Categorcal varable Z wth 4 categores. Z C C2 C3 A 0 0 B 0 0 C 0 0 D Compare Each wth Ffth Example: Categorcal varable Z wth 5 categores. Z C C2 C3 C4 A B C D E Compare Each wth Last Example: Categorcal varable Z wth 4 categores. Z C C2 C3 A 0 0 B 0 0 C 0 0 D Default Reference Value Ths opton specfes the default reference value to be used when automatcally generatng ndcator varables durng the processng of selected categorcal ndependent varables. The reference value s often the baselne, and the other values are compared to t. The choces are Frst Value after Sortng Ffth Value after Sortng Use the frst (through ffth) value n alpha-numerc sorted order as the reference value. Last Value after Sortng Use the last value n alpha-numerc sorted order as the reference value. Frequences Ths s an optonal varable contanng the frequency (observaton count) for each row. Usually, you would leave ths opton blank and let each row receve the default frequency of one. If your data have already been summarzed, ths opton lets you specfy how many actual rows each physcal row represents. Regresson Model Terms Ths opton specfes whch terms (terms, powers, cross-products, and nteractons) are ncluded n the regresson model. For a straght-forward regresson model, select -Way NCSS, LLC. All Rghts Reserved.

14 NCSS Statstcal Software The optons are Posson Regresson Up to -Way Ths opton generates a model n whch each varable s represented by a sngle model term. No crossproducts, nteractons, or powers are added. Use ths opton when you want to use the varables you have specfed, but you do not want to generate other terms. Ths s the opton to select when you want to analyze the ndependent varables specfed wthout addng any other terms. For example, f you have three ndependent varables A, B, and C, ths would generate the model: A + B + C Up to 2-Way Ths opton specfes that all ndvdual varables, two-way nteractons, and squares of numerc varables are ncluded n the model. For example, f you have three numerc varables A, B, and C, ths would generate the model: A + B + C + A*B + A*C + B*C + A*A + B*B + C*C On the other hand, f you have three categorcal varables A, B, and C, ths would generate the model: A + B + C + A*B + A*C + B*C Up to 3-Way All ndvdual varables, two-way nteractons, three-way nteractons, squares of numerc varables, and cubes of numerc varables are ncluded n the model. For example, f you have three numerc, ndependent varables A, B, and C, ths would generate the model: A + B + C + A*B + A*C + B*C + A*B*C + A*A + B*B + C*C + A*A*B + A*A*C + B*B*C +A*C*C + B*C*C On the other hand, f you have three categorcal varables A, B, and C, ths would generate the model: A + B + C + A*B + A*C + B*C + A*B*C Up to 4-Way All ndvdual varables, two-way nteractons, three-way nteractons, and four-way nteractons are ncluded n the model. Also ncluded would be squares, cubes, and quartcs of numerc varables and ther crossproducts. For example, f you have four categorcal varables A, B, C, and D, ths would generate the model: A + B + C + D + A*B + A*C + A*D + B*C + B*D + C*D + A*B*C + A*B*D + A*C*D + B*C*D + A*B*C*D Interacton Manly used for categorcal varables. A saturated model (all terms and ther nteractons) s generated. Ths requres a dataset wth no mssng categorcal-varable combnatons (you can have unequal numbers of observatons for each combnaton of the categorcal varables). No squares, cubes, etc. are generated. For example, f you have three ndependent varables A, B, and C, ths would generate the model: A + B + C + A*B + A*C + B*C + A*B*C Note that the dscusson of the Custom Model opton dscusses the nterpretaton of ths model. Custom Model The model specfed n the Custom Model box s used NCSS, LLC. All Rghts Reserved.

15 NCSS Statstcal Software Posson Regresson Remove Intercept Unchecked ndcates that the ntercept term, β 0, s to be ncluded n the regresson. Checked ndcates that the ntercept should be omtted from the regresson model. Note that deletng the ntercept dstorts most of the dagnostc statstcs (R 2, etc.). In most stuatons, you should nclude the ntercept n the model. Replace Custom Model wth Prevew Model (button) When ths button s pressed, the Custom Model s cleared and a copy of the Prevew model s stored n the Custom Model. You can then edt ths Custom Model as desred. Maxmum Order of Custom Terms Ths opton specfes that maxmum number of varables that can occur n an nteracton (or cross-product) term n a custom model. For example, A*B*C s a thrd order nteracton term and f ths opton were set to 2, the A*B*C term would not be ncluded n the model. Ths opton s partcularly useful when used wth the bar notaton of a custom model to allow a smple way to remove unwanted hgh-order nteractons. Custom Model Ths optons specfes a custom model. It s only used when the Terms opton s set to Custom. A custom model specfes the terms (sngle varables and nteractons) that are to be kept n the model. Interactons An nteracton expresses the combned relatonshp between two or more varables and the dependent varable by creatng a new varable that s the product of the varables. The nteracton between two numerc varables s generated by multplyng them. The nteracton between to categorcal varables s generated by multplyng each par of ndcator varables. The nteracton between a numerc varable and a categorcal varable s created by generatng all products between the numerc varable and the ndcator varables generated from the categorcal varable. Syntax A model s wrtten by lstng one or more terms. The terms are separated by a blank or plus sgn. Terms nclude varables and nteractons. Specfy regular varables (man effects) by enterng the varable names. Specfy nteractons by lstng each varable n the nteracton separated by an astersk (*), such as Frut*Nuts or A*B*C. You can use the bar ( ) symbol as a shorthand technque for specfyng many nteractons quckly. When several varables are separated by bars, all of ther nteractons are generated. For example, A B C s nterpreted as A + B + C + A*B + A*C + B*C + A*B*C. You can use parentheses. For example, A*(B+C) s nterpreted as A*B + A*C. Some examples wll help to ndcate how the model syntax works: A B = A + B + A*B A B A*A B*B = A + B + A*B + A*A + B*B Note that you should only repeat numerc varable. That s, A*A s vald for a numerc varable, but not for a categorcal varable. A A B B (Max Term Order=2) = A + B + A*A + A*B + B*B A B C = A + B + C + A*B + A*C + B*C + A*B*C (A + B)*(C + D) = A*C + A*D + B*C + B*D (A + B) C = (A + B) + C + (A + B)*C = A + B + C + A*C + B*C NCSS, LLC. All Rghts Reserved.

16 NCSS Statstcal Software Subset Selecton Posson Regresson Search Method Ths opton specfes the subset selecton algorthm used to reduce the number of ndependent varables that used n the regresson model. Note that snce the soluton algorthm s teratve, the selecton process can be very tme consumng. The Forward algorthm s much qucker than the Forward wth Swtchng algorthm, but the Forward algorthm does not usually fnd as good of a model. Also note that n the case of categorcal ndependent varables, the algorthm searches among the orgnal categorcal varables, not among the generated ndvdual bnary varables. That s, ether all bnary varables assocated wth a partcular categorcal varable are ncluded or not they are not consdered ndvdually. Herarchcal models are such that f an nteracton s n the model, so are the terms that can be derved from t. For example, f A*B*C s n the model, so are A, B, C, A*B, A*C, and B*C. Statstcans usually adopt herarchcal models rather than non-herarchcal models. The subset selecton procedure can be made to consder only herarchcal models durng ts search. The subset selecton optons are: None No Search s Conducted No subset selecton s attempted. All specfed ndependent varables are used n the regresson equaton. (Herarchcal) Forward Wth ths algorthm, the term wth the largest log lkelhood s entered nto the model. Next, the term that ncreases the log lkelhood the most s added. Ths selecton s contnued untl all the terms have been entered or untl the maxmum subset sze has been reach. If herarchcal models are selected, only those terms that wll keep the model herarchcal are canddates for selecton. For example, the nteracton term A*B wll not be consdered unless both A and B are already n the model. When usng ths algorthm, you must make one run that allows a large number of terms to fnd the approprate number of terms. Next, a second run s made n whch you decrease the maxmum terms n the subset to the number after whch the log lkelhood does not change sgnfcantly. (Herarchcal) Forward wth Swtchng Ths algorthm s smlar to the Forward algorthm descrbed above. The term wth the largest log lkelhood s entered nto the regresson model. The term whch ncreases the log lkelhood the most when combned wth the frst term s entered next. Now, each term n the current model s removed and the rest of the terms are checked to determne f, when they are used nstead, the lkelhood functon s ncreased. If a term can be found by ths swtchng process, the swtch s made and the whole swtchng operaton s begun agan. The algorthm contnues untl no term can be found that mproves the lkelhood. Ths model then becomes the best two-term model. Next, the subset sze s ncreased by one, the best thrd term s entered nto the model, and the swtchng process s repeated. Ths process s repeated untl the maxmum subset sze s reached. Hence, ths model fnds the optmum subset for each subset sze. You must make one run to fnd an approprate subset sze by lookng at the change n the log lkelhood. You then reset the maxmum subset sze to ths value and rerun the analyss. If herarchcal models are selected, only those terms that wll keep the model herarchcal are canddates for addton or deleton. For example, the nteracton term A*B wll not be consdered unless both A and B are already n the model. Lkewse, the term A cannot be removed from a model that contans A*B NCSS, LLC. All Rghts Reserved.

17 NCSS Statstcal Software Posson Regresson Stop search when number of terms reaches Once ths number of terms has been entered nto the model, the subset selecton algorthm s termnated. Often you wll have to run the procedure twce to fnd an approprate value. You would set ths value hgh for the frst run and then reset t approprately for the second run, dependng upon the values of the log lkelhood. Note that the ntercept s counted n ths number. Iteratons Tab Estmaton Optons The followng optons are used durng the lkelhood maxmzaton process. Maxmum Iteratons Specfes the maxmum number of teratons allowed durng the teraton procedure. If ths number s reached, the procedure s termnated prematurely. Typcally, the maxmum lkelhood procedure converges n fve or sx teratons, so a value of twenty here should be ample. Convergence Zero Ths opton specfes the convergence target for the maxmum lkelhood estmaton procedure. When all of the maxmum lkelhood equatons are less than ths amount, the algorthm s assumed to have converged. In theory, all of the equatons should be zero. However, about the best that can be acheved s E-3, so you should set ths value to a number a lttle larger than ths such as the default of E-9. The actual value can be found by lookng at the Maxmum Convergence value on the Run Summary report. Reports Tab The followng optons control whch reports are dsplayed. Varance Adjustment Use Dsperson Ph n SE s Indcate whether to use the Ph multpler n the calculaton of the standard errors of the regresson coeffcents. The Posson model assumes that the mean and varance are dentcal. Usually, the varance s larger than the mean (called overdsperson). A correcton can be appled to the standard errors by multplyng them by the Ph coeffcent. Note that ths correcton wll not change the estmated regresson coeffcents. Alpha Alpha Level Alpha s the sgnfcance level used n the hypothess tests. One mnus alpha s the confdence level of the confdence ntervals. A value of 0.05 s most commonly used. Ths corresponds to a chance of error of n 20. You should not be afrad to use other values snce 0.05 became popular n pre-computer days when t was the only value avalable. Typcal values range from 0.00 to NCSS, LLC. All Rghts Reserved.

18 NCSS Statstcal Software Select Reports Summares Posson Regresson Run Summary... Means Each of these optons specfes whether the ndcated report s calculated and dsplayed. Select Reports Subset Selecton Subset Selecton - Summary and Subset Selecton - Detal Indcate whether to dsplay these subset selecton reports. Select Reports Estmaton Regresson Coeffcents... Rate Coeffcents Indcate whether to dsplay these estmaton reports. Select Reports Goodness-of-Ft Lack-of-Ft Tests... Log-Lkelhood and R-Squared Indcate whether to dsplay these model goodness-of-ft reports. Select Reports Row-by-Row Lsts Resduals... Incdence Indcate whether to dsplay these lst reports. Note that snce these reports provde results for each row, they may be too long for normal use when requested on large databases. Incdence Counts Up to fve ncdence counts may be entered. The probabltes of these counts under the Posson regresson model wll be dsplayed on the Incdence Report. These values must be non-negatve ntegers. Exposure Value Specfy the exposure (tme, space, dstance, volume, etc.) value to be used as a multpler on the Incdence Report. All tems on that report are scaled to ths amount. For example, f your data was scaled n terms of events per month but you want the Incdence report scaled to events per year, you would enter 2 here. Report Optons Tab These optons control format of the reports. Varable Labels Varable Names Ths opton lets you select whether to dsplay only varable names, varable labels, or both NCSS, LLC. All Rghts Reserved.

19 NCSS Statstcal Software Posson Regresson Stagger label and output f label length s The names of the ndcator varables can be too long to ft n the space provded. If the name contans more characters than the number specfed here, only the name s shown on the frst lne of the report and the rest of the output s placed on the next lne. Enter when you want the each varable s results prnted on two lnes. Enter 00 when you want each varable s results prnted on a sngle lne. Decmal Places Precson Specfes whether unformatted numbers (desgnated as decmal places = All ) are dsplayed as sngle (7-dgt) or double (3-dgt) precson numbers n the output. All calculatons are performed n double precson regardless of the Precson selected here. Sngle Unformatted numbers are dsplayed wth 7-dgts. Ths s the default settng. All reports have been formatted for sngle precson. Double Unformatted numbers are dsplayed wth 3-dgts. Ths opton s most often used when the extremely accurate results are needed for further calculaton. For example, double precson mght be used when you are gong to use the Multple Regresson model n a transformaton. Double Precson Format Msalgnment Double precson numbers requre more space than s avalable n the output columns, causng column algnment problems. The double precson opton s for those nstances when accuracy s more mportant than format algnment. Comments. Ths opton does not affect formatted numbers such as probablty levels. 2. Ths opton only nfluences the format of the numbers as they presented n the output. All calculatons are performed n double precson regardless of the Precson selected here. Y... Ch-Square Decmals Specfy the number of dgts after the decmal pont to dsplay on the output of values of ths type. Note that ths opton n no way nfluences the accuracy wth whch the calculatons are done. Enter All to dsplay all dgts avalable. The number of dgts dsplayed by ths opton s controlled by whether the Precson opton s Sngle or Double. Plots Tab These optons control the attrbutes of the varous plots. Select Plots Incdence (Y/T) vs X Plot... Resd vs X Plot Indcate whether to dsplay these plots. Clck the plot format button to change the plot settngs NCSS, LLC. All Rghts Reserved.

20 NCSS Statstcal Software Posson Regresson Edt Durng Run Ths s the small check-box n the upper rght-hand corner of the format button. If checked, the graphcs format wndow for ths plot wll be dsplayed whle the procedure s runnng so that you can format t wth the actual data. Plot Optons Resdual Plotted Ths opton specfes whch of the fve types of resduals are shown on the resdual plots. Storage Tab These optons let you specfy f, and where on the dataset, varous statstcs are stored. Warnng: Any data already n these columns are replaced by the new data. Be careful not to specfy columns that contan mportant data. Data Storage Optons Storage Opton Ths opton controls whether the values ndcated below are stored on the dataset when the procedure s run. Do not store data No data are stored even f they are checked. Store n empty columns only The values are stored n empty columns only. Columns contanng data are not used for data storage, so no data can be lost. Store n desgnated columns Begnnng at the Store Frst Item In column, the values are stored n ths column and those to the rght. If a column contans data, the data are replaced by the storage values. Care must be used wth ths opton because t cannot be undone. Store Frst Item In The frst tem s stored n ths column. Each addtonal tem that s checked s stored n the columns mmedately to the rght of ths column. Leave ths value blank f you want the data storage to begn n the frst blank column on the rght-hand sde of the data. Warnng: any exstng data n these columns s automatcally replaced, so be careful.. Data Storage Optons Select Items to Store Expanded X Values... Covarance Matrx Indcated whether to store these row-by-row values, begnnng at the column ndcated by the Store Frst Item In opton. Note that several of these values nclude a dfferent value for each group and so they requre several columns when they are stored. Expanded X Values Ths opton refers to the expermental desgn matrx. They nclude all bnary and nteracton varables generated NCSS, LLC. All Rghts Reserved.

21 NCSS Statstcal Software Posson Regresson Example Posson Regresson usng a Dataset wth Indcator Varables Ths secton presents several examples. In the frst example, the data shown earler n the Data Structure secton and found n the Koch36 dataset wll be analyzed. Koch et. al. (986) presented ths dataset. It contans the number of new melanoma cases n among whte males n two areas for varous age groups. The sze of the estmated populaton at rsk s gven n the varable Populaton. Ths dataset s nstructve because t shows how easly categorcal varables are dealt wth. In ths example, two categorcal varables, Area and AgeGroup, wll be ncluded n the regresson model. The dataset can also be used to valdate the program snce the results are gven n Koch (986). You may follow along here by makng the approprate entres or load the completed template Example by clckng on Open Example Template from the Fle menu of the Posson Regresson wndow. Open the Koch36 dataset. From the Fle menu of the NCSS Data wndow, select Open Example Data. Clck on the fle Koch36.NCSS. Clck Open. 2 Open the Posson Regresson wndow. Usng the Analyss menu or the Procedure Navgator, fnd and select the Posson Regresson procedure. On the menus, select Fle, then New Template. Ths wll fll the procedure wth the default template. 3 Specfy the varables. On the Posson Regresson wndow, select the Varables, Model tab. Double-clck n the Dependent Y box. Ths wll brng up the varable selecton wndow. Select Melanoma from the lst of varables and clck Ok. Melanoma wll appear n the Dependent Y box. Double-clck n the T: Exposure Varable box. Select Populaton from the lst of varables and clck Ok. Double-clck n the Categorcal X s box. Enter Area(0) AgeGroup(<35) n the Categorcal X s box. The values n parentheses gve the reference value for each varable. The rest of ths panel can be left at the default values. 4 Specfy the model. Set the Terms opton to -Way. Set the Subset Selecton opton to None. 5 Specfy the reports. Select the Reports tab. Check all of the reports and plots. Normally, you would not want all of them, but we specfy them now for documentaton purposes. Set the Incdence Counts to Set the Exposure Value to Specfy the decmals. Select the Report Optons tab. Set the number of decmal places for Probablty to 6. 7 Run the procedure. From the Run menu, select Run Procedure. Alternatvely, just clck the green Run button NCSS, LLC. All Rghts Reserved.

22 NCSS Statstcal Software Run Summary Posson Regresson Item Value Item Value Dependent Varable Melanoma Rows Used 2 Exposure Varable Populaton Sum of Frequences 2 Frequency Varable None Iteratons 5 Ind. Var's Avalable 2 Convergence Zero E-09 No. of X's n Model 6 Maxmum Convergence E-2 Pseudo R² Dsperson Ph.2230 Fnal Lkelhood Ph was not used to correct standard errors. Subset Method None Ths report provdes several detals about the data and the MLE algorthm. Dependent, Exposure, and Frequency Varables These varables are lsted to provde a record of the varables that were analyzed. Ind. Var s Avalable Ths s the number of ndependent varables that you have selected. No. of X s n Model Ths s the number of actual X-varables generated from the terms n the model that was used n the analyss. Pseudo R 2 Ths s the generalzaton of regular R 2 n multple regresson. Ths value s dscussed n detal n the Techncal Detals secton of the chapter. Its formula s R 2 LL = LL ft max LL0 LL Fnal Lkelhood Ths s the value of the log lkelhood that was acheved for ths run. Subset Method Ths s the type of subset selecton that was run. Rows Used Ths s the number of rows used by the estmaton algorthm. Rows wth mssng values and fltered rows are not ncluded. Always check ths value to make sure that you are analyzng all of the data you ntended to. Sum of Frequences Ths s the number of observatons used by the estmaton algorthm. If you specfed a Frequency Varable, ths wll be greater than the number of rows. If not, they wll be equal. Iteratons Ths s number of teratons used by the estmaton algorthm. Usually, the algorthm wll termnate n fve or sx teratons. Convergence Zero The estmaton algorthm contnues untl all of the lkelhood equatons are close to zero. Ths s zero to the algorthm. When the maxmum convergence value s less than ths amount, the algorthm has converged. Compare ths value to the Maxmum Convergence value. Maxmum Convergence The estmaton algorthm contnues untl all of the lkelhood equatons are close to zero. Ths s largest value of all of these equatons. It should be close to zero or the algorthm was termnated before t had converged NCSS, LLC. All Rghts Reserved.

23 NCSS Statstcal Software Posson Regresson Dsperson Ph Ths lne gves the estmated value of the dsperson ph. It also ndcates whether ph was used to adjust the standard errors of the regresson coeffcents and the predcted values. Model Summary Model Error Log Pseudo Model DF DF Lkelhood Devance AIC R 2 Intercept Model Maxmum Ths report s analogous to the analyss of varance table. It summarzes the goodness of ft of the model. Model Ths s the term(s) that are reported about on ths row of the report. Note that the model lne ncludes the ntercept. Model DF Ths s the number of varables n the model. Error DF Ths s the number of observatons mnus the number of varables. Log Lkelhood Ths s the value of the log-lkelhood functon for the ntercept only model, the chosen model, and the saturated model that fts the data perfectly. By comparng these values, you obtan an understandng of how well you model fts the data. Devance The devance s the generalzaton of the sum of squares n regular multple regresson. It measures the dscrepancy between the ftted values and the data. AIC Ths s Akake s nformaton crteron (AIC). It s equal to the devance plus twce the number of parameters n the model. It combnes a measure of the dscrepancy between the ftted values and the data (the devance) wth a measure of the smplcty of the model (twce the number of parameters). It has been shown that usng AIC to compare competng models wth dfferent numbers of parameters amounts to selectng the model wth the mnmum estmate of the mean squared error of predcton. Pseudo R 2 Ths s the generalzaton of regular R 2 n multple regresson. Ths value s dscussed n detal n the Techncal Detals secton of the chapter. Its formula s 2 LL R = LL ft max LL0 LL NCSS, LLC. All Rghts Reserved.

24 NCSS Statstcal Software Means Report Posson Regresson Varable Mean Mnmum Maxmum Melanoma Populaton Ths report gves the mean, mnmum, and maxmum for each of the numerc varables n the analyss. Use t to check for obvous data errors. Regresson Coeffcents Secton Regresson Standard Wald's Lower 95.0% Upper 95.0% Independent Coeffcent Error Ch² Prob Confdence Confdence Varable b() Sb() H0: β=0 Level Lmt Lmt Intercept (Area=) (AgeGroup="35-44") (AgeGroup="45-54") (AgeGroup="54-64") (AgeGroup="65-74") (AgeGroup=">74") Dsperson Ph.2230 Estmated Posson Regresson Model Melanoma = Exp( *(Area=) *(AgeGroup="35-44") *(AgeGroup="45-54") *(AgeGroup="54-64") *(AgeGroup="65-74") *(AgeGroup=">74") ) Ths report provdes the estmated regresson model and assocated statstcs. It provdes the man results of the analyss. Valdaton Koch (986) gves the followng estmates and standard errors. Independent ML Standard Varable Estmate Error Intercept Area AG AG AG AG AG As you can see, these results match those provded by NCSS exactly valdatng our algorthms. These results were also valdated usng SAS. Independent Varable Ths tem provdes the name of the ndependent varable shown on ths lne of the report. The Intercept refers to the optonal constant term. The Dsperson Ph s the estmated value of the ph coeffcent. Note that whether a lne s skpped after the name of the ndependent varable s dsplayed s controlled by the Stagger label and output f label length s opton n the Format tab NCSS, LLC. All Rghts Reserved.

25 NCSS Statstcal Software Posson Regresson Regresson Coeffcent These are the maxmum-lkelhood estmates of the regresson coeffcents, b, b 2,, b k. Ther drect nterpretaton s dffcult because the formula for the predcted value nvolves the exponental functon. Standard Error These are the asymptotc standard errors of the regresson coeffcents, the s b. The estmate the precson of the regresson coeffcent. The standard errors are the square roots of the dagonal elements of ths covarance matrx. The covarance matrx s obtaned by nvertng the observed nformaton matrx evaluated at the maxmum lkelhood estmates. If you Use Dsperson Ph opton, the corrected standard error s shown. Ths s found by multplyng the smple standard error by the square root of ph. That s, the value dsplayed s where s b s = s b b φ Wald s Ch 2 H0: β=0 Ths s the one degree of freedom ch-square statstc for testng the null hypothess that β = 0 aganst the alternatve that β 0. The ch-square value s called a Wald statstc. Ths test has been found to follow the chsquare dstrbuton only n large samples. The test s calculated usng 2 b χ = s b Prob Level The probablty of obtanng a ch-square value greater than the above. Ths s the sgnfcance level of the test. If ths value s less than some predefned alpha level, say 0.05, the varable s sad to be statstcally sgnfcant. Lower and Upper Confdence Lmts These provde a large-sample confdence nterval for the values of the coeffcents. The wdth of the confdence nterval provdes you wth a sense of how precse the regresson coeffcents are. Also, f the confdence nterval ncludes zero, the varable s not statstcally sgnfcant. The formula for the calculaton of the confdence nterval s 2 b ± z s α / 2 where α s the confdence coeffcent of the confdence nterval and z s the approprate value from the standard normal dstrbuton. Dsperson Ph Ths s the estmate of the overdsperson correcton multpler, ph. Remember that n the Posson model the mean and the varance are equal. In practce, the data almost always reject ths restrcton. Usually, the varance s greater than the mean a stuaton called overdsperson. The ncrease n varance s represented n the model by a constant multple of the varance-covarance matrx. That s, we use b where φ s estmated usng V ˆ φ = = n ˆ φ µ β = n n k = x x ( ˆ µ ) y ˆ µ NCSS, LLC. All Rghts Reserved.