STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons... 10 Confdence Intervals... 11 Correlaton Matrx... 11 Unusual Resduals... 12 Resdual Plots... 13 Influental Ponts... 14 Save Results... 16 Calculatons... 16 Summary The Negatve Bnomal Regresson procedure s desgned to ft a regresson model n whch the dependent varable Y conssts of counts. The ftted regresson model relates Y to one or more predctor varables X, whch may be ether quanttatve or categorcal. The procedure fts a model usng ether maxmum lkelhood or weghted least squares. Stepwse selecton of varables s an opton. Lkelhood rato tests are performed to test the sgnfcance of the model coeffcents. The ftted model may be plotted and predctons generated from t. Unusual resduals are dentfed and plotted. Ths procedure s smlar to the Posson Regresson procedure, except that the condtonal varance of Y s allowed to be greater than the mean. It s thus useful for counts that are overdspersed compared to those from a Posson process. Sample StatFolo: Negbn reg.sgp 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 1
STATGRAPHICS Rev. 9/16/2013 Sample Data The fle crabs.sgd contans a set of data from a study of horseshoe crabs, reported by Agrest (2002). The data consst of nformaton on n = 173 female horseshoe crabs. A porton of the data s shown below: Satelltes Wdth 8 28.3 0 22.5 9 26 0 24.8 4 26 0 23.8 0 26.5 0 24.7 0 23.7 0 25.6 It was desred to relate the number of Satelltes (male crabs resdng nearby) to the carapace Wdth of the female crabs. 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 2
Data Input The data nput dalog box requests nformaton about the nput varables: STATGRAPHICS Rev. 9/16/2013 Dependent Varable: a numerc varable contanng the n values of dependent varable y. Y must consst of non-negatve nteger counts. (Sample Szes): optonal sample szes t correspondng to each count. If not specfed, all t are set equal to 1. Quanttatve Factors: numerc columns contanng the values of any quanttatve factors to be ncluded n the model. Categorcal Factors: numerc or non-numerc columns contanng the levels of any categorcal factors to be ncluded n the model. Select: subset selecton. Statstcal Model The statstcal model assumed for the data s that the values of the dependent varable Y follow a negatve bnomal dstrbuton of the form 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 3
p Y 1 1 ( Y ) Y 1 1 ( 1) ( ) 1 1 Y STATGRAPHICS Rev. 9/16/2013, > 0, 0 (1) where the mean s the product of, the rate at whch events occurs, and the samplng perod t accordng to E(Y) = = t (2) The varance of Y s gven by Var(Y) = (3) If = 0, the negatve bnomal dstrbuton reduces to the Posson dstrbuton. It s further assumed that the rate s related to the predctor varables through a log-lnear lnk functon of the form log 0 1X1 2 X 2... k X k (4) Analyss Summary The Analyss Summary dsplays a table showng the estmated model and tests of sgnfcance for the model coeffcents. Typcal output s shown below: Negatve Bnomal Regresson - Satelltes Dependent varable: Satelltes Factors: Wdth Number of observatons: 173 Estmated Regresson Model (Maxmum Lkelhood) Standard Estmated Parameter Estmate Error Rate Rato CONSTANT -4.75395 0.267509 Wdth 0.22032 0.00940901 1.24648 Alpha 0.566406 Analyss of Devance Source Devance Df P-Value Model 24.0164 1 0.0000 Resdual 275.024 171 0.0000 Total (corr.) 299.04 172 Percentage of devance explaned by model = 8.03116 Adjusted percentage = 6.69355 Lkelhood Rato Tests Factor Ch-Squared Df P-Value Wdth 24.0164 1 0.0000 Alpha 195.893 1 0.0000 Resdual Analyss Estmaton n 173 MSE 165.698 MAE 3.27878 MAPE ME -1.3307 MPE Valdaton 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 4
STATGRAPHICS Rev. 9/16/2013 The output ncludes: Data Summary: a summary of the nput data. Estmated Regresson Model: estmates of the coeffcents n the regresson model, wth standard errors and estmated rate ratos. The rate ratos are calculated from the model coeffcents ˆ j by rate rato = ˆ exp (5) j The rate rato represents the percentage ncrease n the rate of events for each unt ncrease n X. Analyss of Devance: decomposton of the devance of the data nto an explaned (Model) component and an unexplaned (Resdual) component. Devance compares the lkelhood functon for a model to the largest value that the lkelhood functon could acheve, n a manner such that a perfect model would have a devance equal to 0. There are 3 lnes n the table: 1. Total (corr.) the devance of a model contanng only a constant term, ( 0 ). 2. Resdual the devance remanng after the model has been ft. 3. Model the reducton n the devance due to the predctor varables, ( 1, 2,, k 0 ), equal to the dfference between the other two components. The P-Value for the Model tests whether the addton of the predctor varables sgnfcantly reduces the devance compared to a model contanng only a constant term. A small P-Value (less than 0.05 f operatng at the 5% sgnfcance level) ndcates that the model has sgnfcantly reduced the devance and s thus a useful for predctor for Y. The P-Value for the Resdual term tests whether there s sgnfcant lack-of-ft,.e., whether a better model may be possble. A small P-value ndcates that sgnfcant devance remans n the resduals, so that a better model mght be possble. Percentage of Devance the percentage of devance explaned by the model, calculated by 0 2 1, 2,..., k 0 R (6) It s smlar to an R-squared statstc n multple regresson, n that t can range from 0% to 100%. An adjusted devance s also computed from R 2 1, 2,..., k 0 2 adj 0 p (7) 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 5
STATGRAPHICS Rev. 9/16/2013 where p equals the number of coeffcents n the ftted model, ncludng the constant term. It s smlar to the adjusted R-squared statstc n that t compensates for the number of varables n the model. Lkelhood Rato Tests a test of sgnfcance for each effect n the ftted model, and for the negatve bnomal parameter. The tests for the effects compare the lkelhood functon of the full model to that of the model n whch only the ndcated effect has been dropped. Small P-values ndcate that the model has been mproved sgnfcantly by the correspondng effect. The test for compares the ftted model aganst a Posson regresson model, correspondng to = 0. A small P-Value for that test ndcates that the data are sgnfcantly overdspersed (the varance s greater than the mean). Resdual Analyss f a subset of the rows n the datasheet have been excluded from the analyss usng the Select feld on the data nput dalog box, the ftted model s used to make predctons of the Y values for those rows. Ths table shows statstcs on the predcton errors, defned by e ˆ (8) y Included are the mean squared error (MSE), the mean absolute error (MAE), the mean absolute percentage error (MAPE), the mean error (ME), and the mean percentage error (MPE). These valdaton statstcs can be compared to the statstcs for the ftted model to determne how well that model predcts observatons outsde of the data used to ft t. The ftted model for the sample data s ˆ exp 4.75395 0.22032Wdth (9) The regresson explans a lttle more than 8% of the devance of a model wth only a constant. The P-value for Wdth s very small, ndcatng that t s a sgnfcant predctor for Satelltes. The mean and varance of Satelltes are gven by: E( Satelltes) ˆ exp 4.75395 0. 22032Wdth (10) Var ( Satelltes) ˆ 1 0.566406 ˆ (11) In addton, the P-value for Alpha s very small, ndcatng that there s sgnfcant overdsperson, so that a Posson model would not be approprate for ths data. 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 6
STATGRAPHICS Rev. 9/16/2013 Analyss Optons Model: order of the model to be ft. Frst order models nclude only man effects. Second order models nclude quadratc effects for quanttatve factors and two-factor nteractons amongst all varables. Include Constant: If ths opton s not checked, the constant term 0 wll be omtted from the model. Ft: specfes whether all ndependent varables specfed on the data nput dalog box should be ncluded n the fnal model, or whether a stepwse selecton of varables should be appled. Stepwse selecton attempts to fnd a parsmonous model that contans only statstcally sgnfcant varables. A Forward Stepwse ft begns wth no varables n the model. A Backward Stepwse ft begns wth all varables n the model. P-to-Enter - In a stepwse ft, varables wll be entered nto the model at a gven step f ther P-values are less than or equal to the P-to-Enter value specfed. P-to-remove - In a stepwse ft, varables wll be removed from the model at a gven step f ther P-values are greater than the P-to-Remove value specfed. Max Steps: maxmum number of steps permtted when dong a stepwse ft. Dsplay: whether to dsplay the results at each step when dong a stepwse ft. Exclude: Press ths button to exclude effects from the model. A dalog box wll be dsplayed: 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 7
Satelltes STATGRAPHICS Rev. 9/16/2013 Double clck on an effect to move t from the Include feld to the Exclude feld or back agan. Plot of Ftted Model The Plot of Ftted Model dsplays the estmated mean rate ˆ ( X ) versus any sngle predctor varable, wth the other varables held constant. 15 Plot of Ftted Model wth 95.0% confdence lmts 12 9 6 3 0 21 24 27 30 33 36 Wdth Confdence lmts for X) are ncluded on the plot. The estmated mean number of satelltes ncreases from a low of approxmately 1 at small wdths to over a dozen at large wdths. 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 8
Pane Optons STATGRAPHICS Rev. 9/16/2013 Factor: select the factor to plot on the horzontal axs. Low and Hgh: specfy the range of values for the selected factor. Hold: values to hold the unselected factors at. Confdence Level: percentage used for the confdence lmts. Set to 0 to suppress the lmts. 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 9
observed STATGRAPHICS Rev. 9/16/2013 Observed Versus Predcted The Observed versus Predcted plot shows the observed values of Y on the vertcal axs and the predcted mean values ˆ on the horzontal axs. 15 12 Plot of Satelltes 9 6 3 0 0 3 6 9 12 15 predcted If the model fts well, the ponts should be randomly scattered around the dagonal lne. Predctons The ftted regresson model may be used to predct the outcome of new samples whose predctor varables are gven. For example, suppose a predcton s desred for a crab wth Wdth = 30.3. A new row could be added to the datasheet wth 30.3 n the Wdth column, but the entry for Satelltes would be left blank. The Predctons pane would then dsplay: Predctons for Satelltes Observed Ftted Lower 95.0% CL Upper 95.0% CL Row Value Value for Predcton for Predcton 174 6.83284 6.44308 7.24619 The table shows the ftted value ˆ, together wth approxmate 95% confdence ntervals. Pane Optons 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 10
STATGRAPHICS Rev. 9/16/2013 Dsplay: dsplay All Values (predctons for all rows n the datasheet), or Forecasts Only (predctons for rows wth mssng values for Y). Confdence Level: percentage used by the confdence ntervals. Confdence Intervals The Confdence Intervals pane shows the potental estmaton error assocated wth each coeffcent n the model, as well as for the rate ratos. 95.0% confdence ntervals for coeffcent estmates Standard Parameter Estmate Error Lower Lmt Upper Lmt CONSTANT -4.75395 0.267509-5.27826-4.22965 Wdth 0.22032 0.00940901 0.201879 0.238761 95.0% confdence ntervals for rate ratos Parameter Estmate Lower Lmt Upper Lmt Wdth 1.24648 1.2237 1.26968 Pane Optons Confdence Level: percentage level for the confdence ntervals. Correlaton Matrx The Correlaton Matrx dsplays estmates of the correlaton between the estmated coeffcents. Correlaton matrx for coeffcent estmates CONSTANT Wdth CONSTANT 1.0000-0.9961 Wdth -0.9961 1.0000 Ths table can be helpful n determnng how well the effects of dfferent ndependent varables have been separated from each other. 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 11
STATGRAPHICS Rev. 9/16/2013 Unusual Resduals Once the model has been ft, t s useful to study the resduals to determne whether any outlers exst that should be removed from the data. The Unusual Resduals pane lsts all observatons that have unusually large resduals. Unusual Resduals for Satelltes Predcted Pearson Devance Row Y Y Resdual Resdual Resdual 3 9.0 2.64948 6.35052 2.47 1.68 15 14.0 2.64948 11.3505 4.41 2.55 34 8.0 2.48003 5.51997 2.26 1.57 56 15.0 4.39778 10.6022 2.71 1.80 121 6.0 1.63176 4.36824 2.47 1.67 131 6.0 1.90386 4.09614 2.06 1.46 134 10.0 2.53527 7.46473 3.00 1.94 146 8.0 2.12558 5.87442 2.71 1.80 149 10.0 1.70527 8.29473 4.53 2.58 The table dsplays: Row the row number n the datasheet. Y the observed value of Y. Predcted Y the ftted value ˆ. Resdual the dfference between the observed and predcted values, defned by e ˆ (12) y Pearson Resdual a standardzed resdual n whch each resdual s dvded by an estmate of ts standard error: r (13) e ˆ 1 ˆ ˆ Devance Resdual a resdual that measures each observaton s contrbuton to the resdual devance: 1 1 1 y ˆ ln y ˆ / ˆ ˆ y d sgn( r ) 2 y ln (14) ˆ The sum of squared devance resduals equals the devance on the Resduals lne of the analyss of devance table. The table ncludes all rows for whch the absolute value of the Pearson resdual s greater than 2.0. The current example shows 9 resduals that exceed 2.5, 2 of whch exceed 3.0. 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 12
autocorrelaton devance resdual STATGRAPHICS Rev. 9/16/2013 Resdual Plots As wth all statstcal models, t s good practce to examne the resduals. The Negatve Bnomal Regresson procedure ncludes varous type of resdual plots, dependng on Pane Optons. Scatterplot versus Predcted Value Ths plot s helpful n examnng very large resduals. 5.1 Resdual Plot 3.1 1.1-0.9-2.9-4.9 0 3 6 9 12 15 predcted Satelltes Resdual Autocorrelatons Ths plot calculates the autocorrelaton between resduals as a functon of the number of rows between them n the datasheet. 1 Resdual Autocorrelatons for Satelltes 0.6 0.2-0.2-0.6-1 0 2 4 6 8 10 12 lag It s only relevant f the data have been collected sequentally. Any bars extendng beyond the probablty lmts would ndcate sgnfcant dependence between resduals separated by the ndcated lag. 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 13
Pane Optons STATGRAPHICS Rev. 9/16/2013 Plot: the type of resduals to plot: 1. Resduals the observed values mnus the ftted values. 2. Studentzed resduals the resduals dvded by ther estmated standard errors. 3. Devance Resduals resduals scaled so that ther sum of squares equals the resdual devance. Type: the type of plot to be created. A Scatterplot s used to test for curvature. A Normal Probablty Plot s used to determne whether the model resduals come from a normal dstrbuton (normalty s not expected n ths procedure). An Autocorrelaton Functon s used to test for dependence between consecutve resduals. Plot Versus: for a Scatterplot, the quantty to plot on the horzontal axs. Number of Lags: for an Autocorrelaton Functon, the maxmum number of lags. For small data sets, the number of lags plotted may be less than ths value. Confdence Level: for an Autocorrelaton Functon, the level used to create the probablty lmts. Influental Ponts In fttng a regresson model, all observatons do not have an equal nfluence on the parameter estmates n the ftted model. Those wth unusual values of the ndependent varables tend to have more nfluence than the others. The Influental Ponts pane dsplays any observatons that have hgh nfluence on the ftted model: 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 14
STATGRAPHICS Rev. 9/16/2013 Influental Ponts for Satelltes Row Leverage 115 0.113515 141 0.393344 142 0.0362742 147 0.0966221 Average leverage of sngle data pont = 0.0115607 The table dsplays all ponts wth hgh leverage. Leverage s a statstc that measures how dstant an observaton s from the mean of all n observatons n the space of the ndependent varables. The hgher the leverage, the greater the mpact of the pont on the ftted values ŷ. Ponts are placed on the lst f ther leverage s more than 3 tmes that of an average data pont. The observaton wth the hghest leverage n the sample data s row #141. It s over 30 tmes the average leverage, whch means t has a large nfluence on the ft. 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 15
STATGRAPHICS Rev. 9/16/2013 Save Results The followng results may be saved to the datasheet: 1. Predcted Values the ftted values ˆ t correspondng to each row of the datasheet. 2. Lower Lmts the lower confdence lmts for ˆ t. 3. Upper Lmts the upper confdence lmts for ˆ t. 4. Resduals the ordnary resduals. 5. Pearson Resduals the standardzed Pearson resduals. 6. Devance Resduals the devance resduals. 7. Leverages the leverages for each row. Calculatons Let = the estmated mean at the settngs of the predctor varables n row. Lkelhood Functon L n 1 1 y 1 y 1 1 1 1 1 y for 0, 0 (15) Devance n 1 1 ( ˆ ) y ln y ln 1 (16) 1 ˆ ˆ y y Leverage h w 1 dag X X WX X (17) p h (18) n 2013 by StatPont Technologes, Inc. Negatve Bnomal Regresson - 16