STAT 3220 Unit 3

During periods of high electricity demand, especially during the hot summer months, the power output from a gas turbine engine can drop dramatically. One way to counter this drop in power is by cooling the inlet air to the gas turbine. An increasingly popular cooling method uses high pressure inlet fogging. The performance of a sample of 67 gas turbines augmented with high pressure inlet fogging was investigated in the Journal of Engineering for Gas Turbines and Power (January 2005). One measure of performance is heat rate (kilojoules per kilowatt per hour). Heat rates for the 67 gas turbines, saved in the gasturbine file. Read some of the engineering applications from this journal here:

Step 1: Collect the Data

Data Compilation: These data were collected from researchers in a reputable journal.
Data Organization: Are the data organized for MLR?- each row contains information about one observation (experimental unit) including all of the explanatory predictors and the response.
Suitability of response variable for regression: View a histogram of response variable. Primarily, it should be continuous with few outliers. Data should not be recorded over time. It is sometimes useful if the response variable is unimodal and symmetric, but that is NOT a requirement. (Further, we will not use “rank” as a response variable in MLR. We need a continuous variable with several values of each response, if the variable is discrete.)

gasturbine<-read.delim("https://raw.githubusercontent.com/kvaranyak4/STAT3220/main/GASTURBINE.txt")
head(gasturbine)

names(gasturbine)

[1] "ENGINE"     "SHAFTS"     "RPM"        "CPRATIO"    "INLET.TEMP"
[6] "EXH.TEMP"   "AIRFLOW"    "POWER"      "HEATRATE"

gasturbine$log.CPRATIO<-log(gasturbine$CPRATIO)
gasturbine$log.AIRFLOW<-log(gasturbine$AIRFLOW)
gasturbine$log.POWER<-log(gasturbine$POWER)

hist(gasturbine$HEATRATE, xlab="Heat Rate", main="Histogram of Heat Rate")

The distribution of the response variable, heat rate, is unimodal and skewed right. It is continuous, so it should still be suitable for regression.

Step 2: Hypothesize Relationship (Exploratory Data Analysis)

Recall (From Unit 2.4)

Most of the variables have high correlations. Exhaust temperature has the weakest correlation.
CPRATIO, POWER and AIRFLOW should have a log transformation because their correlations with HEATRATE were much higher, indicating a stronger linear form with that transformation.
Engine type appears to have a different average heat rate for the advanced group.
Note: Shafts=3 only has two observations. Further, we would not be able to fit an interaction between Engine and Shaft because some level combinations do not have any observations.
There is evidence of severe multicollinearity because all VIFs are MUCH greater than 10 and the average VIF is MUCH greater than 3 at 393.2362.

#updated model: REMOVE LOG.POWER
gasmod3<-lm(HEATRATE~RPM+INLET.TEMP+EXH.TEMP+log.AIRFLOW,data=gasturbine)
gasmod3vif<-round(vif(gasmod3),3)
gasmod3vif

        RPM  INLET.TEMP    EXH.TEMP log.AIRFLOW 
     10.057       3.575       3.334      10.683

mean(gasmod3vif)

[1] 6.91225

#updated model: REMOVE LOG.AIRFLOW
gasmod4<-lm(HEATRATE~RPM+INLET.TEMP+EXH.TEMP+log.POWER,data=gasturbine)
gasmod4vif<-round(vif(gasmod4),3)
gasmod4vif

       RPM INLET.TEMP   EXH.TEMP  log.POWER 
     9.777      3.676      3.271     12.128

mean(gasmod4vif)

[1] 7.213

#updated model: REMOVE BOTH
gasmod5<-lm(HEATRATE~RPM+INLET.TEMP+EXH.TEMP,data=gasturbine)
gasmod5vif<-round(vif(gasmod5),3)
gasmod5vif

       RPM INLET.TEMP   EXH.TEMP 
     1.727      3.570      2.551

mean(gasmod5vif)

[1] 2.616

It appears the model without power and airflow have resolved the issue of severe multicollinearity. HOWEVER, we may have also decided to remove a combination of RPM and power/airflow, as that also had a strong relationship. As a researcher, or with your client, you may discuss which of those three variables would be most important to analyze moving forward. Here, we will remove the transformations as we want the most least complex model.

Step 3: Estimate the Model Parameters

Stage 1: Quant

Starting with the subset of predictors: RPM, INLET.TEMP,and EXH.TEMP

#updated model
gasmod5<-lm(HEATRATE~RPM+INLET.TEMP+EXH.TEMP,data=gasturbine)

summary(gasmod5)


Call:
lm(formula = HEATRATE ~ RPM + INLET.TEMP + EXH.TEMP, data = gasturbine)

Residuals:
    Min      1Q  Median      3Q     Max 
-1025.8  -297.9  -115.3   225.8  1425.1 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.436e+04  7.333e+02  19.582  < 2e-16 ***
RPM          1.051e-01  1.071e-02   9.818 2.55e-14 ***
INLET.TEMP  -9.223e+00  7.869e-01 -11.721  < 2e-16 ***
EXH.TEMP     1.243e+01  2.071e+00   6.000 1.06e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 465 on 63 degrees of freedom
Multiple R-squared:  0.9189,    Adjusted R-squared:  0.915 
F-statistic: 237.9 on 3 and 63 DF,  p-value: < 2.2e-16

Stage 2: Qual

Try adding qualitative variable Engine

#updated model
gasmod6<-lm(HEATRATE~RPM+INLET.TEMP+EXH.TEMP+ENGINE,data=gasturbine)

summary(gasmod6)


Call:
lm(formula = HEATRATE ~ RPM + INLET.TEMP + EXH.TEMP + ENGINE, 
    data = gasturbine)

Residuals:
     Min       1Q   Median       3Q      Max 
-1030.37  -247.21   -70.62   258.72  1400.82 

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)        1.607e+04  1.262e+03  12.732  < 2e-16 ***
RPM                1.077e-01  1.198e-02   8.991 8.85e-13 ***
INLET.TEMP        -9.899e+00  9.374e-01 -10.561 2.12e-15 ***
EXH.TEMP           1.116e+01  2.202e+00   5.068 4.00e-06 ***
ENGINEAeroderiv   -4.404e+02  2.870e+02  -1.534    0.130    
ENGINETraditional -3.667e+02  2.233e+02  -1.642    0.106    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 461.2 on 61 degrees of freedom
Multiple R-squared:  0.9227,    Adjusted R-squared:  0.9164 
F-statistic: 145.6 on 5 and 61 DF,  p-value: < 2.2e-16

Most important predictors

Perform nested F test to determine if Engine type is significant.

#anova(reduced,complete)
anova(gasmod5,gasmod6)

Hypotheses:
- \(H_0: \beta_4=\beta_5=0\) (the engine type does not contribute to predicting heat rate)
- \(H_a:\beta_4, \beta_5 \neq 0\) (the engine type contributes to predicting heat rate)
Distribution of test statistic: F 2 with 61 DF
Test Statistic: F=1.5137
Pvalue: 0.2282
Decision: 0.2282>0.05 -> FAIL TO REJECT H0
Conclusion: The engine type is not significant at predicting heat rate. We will remove both dummy variables in the model and not test them individually.

Qual X Quant

For the sake of this example, we will not consider any further testing.

Step 4: Specify the distribution of the errors and find the estimate of the variance

We have covered this individually and will come back to it in a complete example at the end of the unit.

Step 5: Evaluate the Utility of the model

We have covered this individually and will come back to it in a complete example at the end of the unit.

Step 6: Check the Model Assumptions (Residual Analysis)

Assumptions

First we will evaluate model assumptions

#There are a few options to view plots for assumptions

#store residuals from the model
gasres<-residuals(gasmod5)
sum(gasres)

[1] 3.552714e-14

mean(gasres)

[1] 5.140995e-16

#Residuals Plots of explanatory variables vs residuals
residualPlots(gasmod5,tests=F)

#Residual vs Fitted and QQ plot
plot(gasmod5, which=c(1,2))

#histogram of residuals
hist(gasres)

Due to rounding errors within R we see the mean and sum of the residuals are not quite zero.

Add your notes for each assumption:

Lack of Fit: Residual Plots
Constant Variance: Residual Plots and Residual vs Fitted
Normality: QQ Plot and Histogram of Residuals
Independence: For the sake of this class we will not explore this assumption other than stating we do not believe the observations are dependent on each other.

Influential Diagnostics

Then we will identify outliers and influential observations.

#Cooks Distance Thresholds
plot(gasmod5,which=4)

#Leverage vs Studentized Residuals
influencePlot(gasmod5,fill=F)

#Deleted Studentized Residuals vs Predicted values
ols_plot_resid_stud_fit(gasmod5)

Label the observations that exceed the threshold for each statistic:

Cook’s Distance: Identifies influential observations. Threshold is 4/n
Leverage (Hat): Identifies outliers in the x direction (extreme values of explanatory variables). Threshold is 2(k+1)/n
Studentizied Residuals: Identifies outliers in the y direction. Threshold is +/- 2
Circles on “influence plot” are indicative of the cook’s distance value
Deleted Studentizied Residuals (R Student): Identifies general outliers and influential observations. Threshold is +/- 2

After identifying the observations that exceed one or more of the thresholds above, we investigate further into their DFFits and DFBetas.

#We can use various functions to store and view these statistics for all observations
# We will not print these outputs for every observation into reports
rstudent(gasmod5)

           1            2            3            4            5            6 
 0.893217621  1.317528848 -0.692830495 -1.629043379 -0.319227791 -0.220252658 
           7            8            9           10           11           12 
-0.233471856  0.374990192 -0.341559467 -0.680104090  3.389504856  2.116744668 
          13           14           15           16           17           18 
-0.253968278  0.011011147  0.920546030 -0.303301225 -0.707442804 -0.947506151 
          19           20           21           22           23           24 
-0.649930306  0.488456104  1.541375696  0.493725647 -0.528770732 -0.649760795 
          25           26           27           28           29           30 
-0.295878983 -0.443466676 -0.856333719 -0.004877388 -0.196240609 -0.795751546 
          31           32           33           34           35           36 
-0.837057024 -2.367042399  0.363745397  0.012398052 -1.128484921  2.749086328 
          37           38           39           40           41           42 
 0.224875075  0.043825526 -1.568397362 -0.516762527  0.288577835 -0.286838938 
          43           44           45           46           47           48 
-0.454237853 -0.520022587  1.285279709 -0.257821207  2.942194529  1.259842609 
          49           50           51           52           53           54 
-0.344977609  0.691549153 -0.326595642  0.658329323  1.297334067  0.046125589 
          55           56           57           58           59           60 
 0.458976459 -0.683477727 -0.209983135 -0.860119314  0.428369068 -1.098019290 
          61           62           63           64           65           66 
 0.548011700 -1.431960144 -0.794961623  0.722757693 -0.538743015  0.683098419 
          67 
-0.452855099

hatvalues(gasmod5)

         1          2          3          4          5          6          7 
0.18485774 0.08148766 0.04852153 0.04180664 0.02953543 0.02090636 0.02675674 
         8          9         10         11         12         13         14 
0.03347980 0.03328687 0.02797014 0.04627741 0.03905134 0.06132261 0.03904983 
        15         16         17         18         19         20         21 
0.09447390 0.03578988 0.04966096 0.05602507 0.03200633 0.04489513 0.03346492 
        22         23         24         25         26         27         28 
0.02275577 0.03188287 0.04122101 0.02458622 0.03941415 0.03917409 0.10819365 
        29         30         31         32         33         34         35 
0.08678043 0.03123228 0.03193504 0.06782971 0.06092269 0.03689212 0.04035201 
        36         37         38         39         40         41         42 
0.14870478 0.05221727 0.06619795 0.04114569 0.03406287 0.04011764 0.04930437 
        43         44         45         46         47         48         49 
0.08149383 0.05111380 0.08925279 0.07719304 0.02446354 0.02446354 0.05725267 
        50         51         52         53         54         55         56 
0.06910879 0.04056331 0.07048539 0.04601525 0.03741620 0.02774057 0.03778892 
        57         58         59         60         61         62         63 
0.04257101 0.04879281 0.06952830 0.03789756 0.20397249 0.18110748 0.04996695 
        64         65         66         67 
0.27295893 0.11306102 0.05760075 0.03264216

cooks.distance(gasmod5)

           1            2            3            4            5            6 
4.537897e-02 3.805605e-02 6.170622e-03 2.820616e-02 7.865760e-04 2.629333e-04 
           7            8            9           10           11           12 
3.803540e-04 1.234573e-03 1.018545e-03 3.356037e-03 1.194756e-01 4.313782e-02 
          13           14           15           16           17           18 
1.069302e-03 1.251617e-06 2.215619e-02 8.661275e-04 6.590471e-03 1.334232e-02 
          19           20           21           22           23           24 
3.524009e-03 2.838050e-03 2.012551e-02 1.436298e-03 2.328622e-03 4.579823e-03 
          25           26           27           28           29           30 
5.597668e-04 2.043391e-03 7.506244e-03 7.331528e-07 9.290596e-04 5.133514e-03 
          31           32           33           34           35           36 
5.806060e-03 9.498435e-02 2.175884e-03 1.495731e-06 1.332918e-02 2.989222e-01 
          37           38           39           40           41           42 
7.071683e-04 3.458752e-05 2.579141e-02 2.381973e-03 8.829767e-04 1.082513e-03 
          43           44           45           46           47           48 
4.635059e-03 3.684404e-03 4.005790e-02 1.411003e-03 4.838904e-02 9.858692e-03 
          49           50           51           52           53           54 
1.832476e-03 8.950190e-03 1.143616e-03 8.290733e-03 2.007800e-02 2.100772e-05 
          55           56           57           58           59           60 
1.521703e-03 4.625639e-03 4.976870e-04 9.526546e-03 3.472962e-03 1.183408e-02 
          61           62           63           64           65           66 
1.945420e-02 1.115140e-01 8.358358e-03 4.940477e-02 9.354983e-03 7.191049e-03 
          67 
1.752125e-03

dffits(gasmod5)

           1            2            3            4            5            6 
 0.425362695  0.392431237 -0.156456917 -0.340273949 -0.055690680 -0.032184608 
           7            8            9           10           11           12 
-0.038711537  0.069792041 -0.063380257 -0.115367354  0.746637455  0.426713445 
          13           14           15           16           17           18 
-0.064912990  0.002219687  0.297338371 -0.058434377 -0.161718462 -0.230830397 
          19           20           21           22           23           24 
-0.118181220  0.105900892  0.286810219  0.075340741 -0.095958207 -0.134726673 
          25           26           27           28           29           30 
-0.046974865 -0.089829495 -0.172910016 -0.001698842 -0.060493984 -0.142879357 
          31           32           33           34           35           36 
-0.152032537 -0.638511006  0.092648067  0.002426514 -0.231404802  1.148975438 
          37           38           39           40           41           42 
 0.052782991  0.011668687 -0.324894209 -0.097041437  0.058995878 -0.065322088 
          43           44           45           46           47           48 
-0.135302132 -0.120693600  0.402355212 -0.074567961  0.465917825  0.199505207 
          49           50           51           52           53           54 
-0.085014131  0.188425741 -0.067153544  0.181286164  0.284925934  0.009093948 
          55           56           57           58           59           60 
 0.077527720 -0.135447627 -0.044278010 -0.194804528  0.117097421 -0.217923939 
          61           62           63           64           65           66 
 0.277403278 -0.673419464 -0.182313250  0.442855356 -0.192349579  0.168880642 
          67 
-0.083186942

dfbetas(gasmod5)

     (Intercept)           RPM    INLET.TEMP      EXH.TEMP
1  -0.2543877689  0.2702354892 -0.0179373762  0.1674385619
2   0.3169951733  0.0722423682  0.0181069528 -0.2216629859
3   0.0350496972 -0.1276702220 -0.0614516574  0.0338096397
4  -0.2663107890 -0.0117487442  0.0237567805  0.1466728252
5   0.0196958361 -0.0239481520  0.0056188036 -0.0177883757
6  -0.0148881596  0.0037126094 -0.0068511914  0.0138466510
7  -0.0191213854 -0.0003158663 -0.0149794844  0.0236436589
8   0.0470740849 -0.0302973389 -0.0012477769 -0.0252372918
9  -0.0283704899  0.0455909083  0.0292700817 -0.0114556375
10 -0.0499094478  0.0732438446  0.0313107132 -0.0032886764
11  0.0146396925  0.3343853805 -0.1771736110  0.1326314944
12  0.1232879232 -0.1253086407 -0.3120523904  0.2018511581
13  0.0372577985 -0.0498266103 -0.0259600958 -0.0006396223
14  0.0017823141 -0.0001860978  0.0001692894 -0.0012330015
15  0.1977245682 -0.2021856920 -0.2216410817  0.0761575306
16 -0.0093013852  0.0388802988  0.0411335435 -0.0331455805
17 -0.0471210539  0.1240764064  0.1202542833 -0.0834658159
18 -0.0494525368  0.1810355233  0.1754822278 -0.1340271108
19 -0.0010742113  0.0801707160  0.0566949872 -0.0570467492
20  0.0008172527  0.0803922302  0.0696819070 -0.0618691555
21  0.1187745030 -0.2061358762 -0.1569317982  0.0785190805
22  0.0428244933 -0.0272943223 -0.0048392793 -0.0189445851
23 -0.0400212720  0.0679344585  0.0488067593 -0.0230029314
24 -0.0217465205  0.0997861868  0.0923538285 -0.0746654188
25 -0.0198922461  0.0257376418  0.0079269807  0.0026016883
26 -0.0537032798  0.0633301465  0.0408688041 -0.0064580429
27 -0.0457276255  0.1323798380  0.1082248136 -0.0759656482
28 -0.0002085654 -0.0003846534  0.0009064725 -0.0006180903
29 -0.0488622090 -0.0085233208  0.0048446371  0.0276434498
30 -0.0313198381 -0.0467032435  0.0332095017 -0.0082556456
31 -0.0309523904 -0.0909612222 -0.0329026308  0.0498176640
32 -0.1931169238 -0.4379105485 -0.4048748978  0.4843855487
33  0.0737598798  0.0073407101  0.0255699763 -0.0682455525
34  0.0017093348  0.0002373894  0.0006738580 -0.0016300107
35 -0.1135493407 -0.1005506429 -0.1155057555  0.1720134599
36  0.4159486523 -0.8478462527 -1.0529590426  0.6883629665
37  0.0268483526 -0.0399948767 -0.0364759545  0.0172814081
38  0.0084380646 -0.0082805629 -0.0065775191  0.0008495385
39 -0.2454907677 -0.0399489002 -0.0450242507  0.1928232646
40  0.0170560223 -0.0339095604 -0.0699575479  0.0463968659
41 -0.0348131223 -0.0146242346 -0.0118366061  0.0358625427
42  0.0407190770  0.0172278269  0.0079654353 -0.0367484298
43  0.0995720234  0.0284920021  0.0215456293 -0.0889537913
44  0.0716272593  0.0386398233  0.0191398725 -0.0696826252
45 -0.1217605776  0.1276654686  0.3324443647 -0.1981042737
46  0.0565912155  0.0012618619 -0.0154921335 -0.0262649022
47 -0.1617510315  0.1304506996  0.2378441134 -0.0829745042
48 -0.0692615120  0.0558587640  0.1018444381 -0.0355295392
49  0.0570234576  0.0198389084  0.0097739121 -0.0496148764
50 -0.1149315241  0.0389499084  0.1092792057 -0.0136862294
51  0.0367650040  0.0105587319 -0.0119780590 -0.0172472463
52 -0.0931367393  0.0387876514  0.1192455431 -0.0365763464
53  0.0118140083  0.0849943750  0.2219197485 -0.1910320807
54 -0.0058482057 -0.0008127183  0.0002289599  0.0040314977
55 -0.0103240501 -0.0063519200  0.0335033940 -0.0176457828
56  0.0660508365  0.0010923255 -0.0527086876 -0.0042019584
57  0.0268198360  0.0039992886 -0.0088200040 -0.0120370028
58  0.0744459021  0.1050025785  0.0891544108 -0.1373239192
59 -0.0692634855  0.0257390261  0.0707605276 -0.0124353092
60  0.1099146775  0.0638837819  0.0032344382 -0.0874225183
61 -0.0201404718  0.1856806933 -0.0271551865  0.0248305889
62  0.2369879424 -0.4289108189  0.0899488620 -0.2068438091
63  0.0208192376 -0.1098873963  0.0117639805 -0.0198152615
64  0.1723990277  0.1463708012  0.3612199728 -0.4209772281
65 -0.1271968460 -0.0118060343 -0.1041312850  0.1680582067
66 -0.0799292230  0.1375459933  0.0898178147 -0.0286093893
67  0.0032007185 -0.0184425594  0.0334306140 -0.0313356072

# Instead we can subset by the observations of interest
dffits(gasmod5)[c(11,36,61,64)]

       11        36        61        64 
0.7466375 1.1489754 0.2774033 0.4428554

summary(dffits(gasmod5))

    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
-0.67342 -0.13501 -0.05569  0.01663  0.11150  1.14898

dfbetas(gasmod5)[c(11,36,61,64),]

   (Intercept)        RPM  INLET.TEMP    EXH.TEMP
11  0.01463969  0.3343854 -0.17717361  0.13263149
36  0.41594865 -0.8478463 -1.05295904  0.68836297
61 -0.02014047  0.1856807 -0.02715519  0.02483059
64  0.17239903  0.1463708  0.36121997 -0.42097723

summary(dfbetas(gasmod5))

  (Intercept)              RPM              INLET.TEMP       
 Min.   :-0.2663108   Min.   :-0.847846   Min.   :-1.052959  
 1st Qu.:-0.0479916   1st Qu.:-0.025621   1st Qu.:-0.016715  
 Median :-0.0002086   Median : 0.003999   Median : 0.007927  
 Mean   : 0.0013664   Mean   :-0.001314   Mean   :-0.001471  
 3rd Qu.: 0.0417718   3rd Qu.: 0.070088   3rd Qu.: 0.052751  
 Max.   : 0.4159486   Max.   : 0.334385   Max.   : 0.361220  
    EXH.TEMP         
 Min.   :-0.4209772  
 1st Qu.:-0.0533308  
 Median :-0.0124353  
 Mean   : 0.0005346  
 3rd Qu.: 0.0242371  
 Max.   : 0.6883630

Next Steps to resolve

Variable transformation: We can directly use a function of the response variable within lm() or add a new variable to the table

# directly transform in the lm() function

gasmod5sqrt<-lm(sqrt(HEATRATE)~RPM+INLET.TEMP+EXH.TEMP,data=gasturbine)
summary(gasmod5sqrt)


Call:
lm(formula = sqrt(HEATRATE) ~ RPM + INLET.TEMP + EXH.TEMP, data = gasturbine)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.4872 -1.3479 -0.5044  0.9321  6.2759 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.218e+02  3.346e+00  36.392  < 2e-16 ***
RPM          4.640e-04  4.886e-05   9.496 9.05e-14 ***
INLET.TEMP  -4.367e-02  3.590e-03 -12.163  < 2e-16 ***
EXH.TEMP     5.706e-02  9.449e-03   6.039 9.12e-08 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.122 on 63 degrees of freedom
Multiple R-squared:  0.9206,    Adjusted R-squared:  0.9168 
F-statistic: 243.4 on 3 and 63 DF,  p-value: < 2.2e-16

# OR Create new variable
gasturbine$sqrty<-sqrt(gasturbine$HEATRATE)

#remember to remove original response heatrate
gasmodsqrt2<-lm(sqrty~RPM+INLET.TEMP+EXH.TEMP,data=gasturbine)
summary(gasmodsqrt2)


Call:
lm(formula = sqrty ~ RPM + INLET.TEMP + EXH.TEMP, data = gasturbine)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.4872 -1.3479 -0.5044  0.9321  6.2759 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.218e+02  3.346e+00  36.392  < 2e-16 ***
RPM          4.640e-04  4.886e-05   9.496 9.05e-14 ***
INLET.TEMP  -4.367e-02  3.590e-03 -12.163  < 2e-16 ***
EXH.TEMP     5.706e-02  9.449e-03   6.039 9.12e-08 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.122 on 63 degrees of freedom
Multiple R-squared:  0.9206,    Adjusted R-squared:  0.9168 
F-statistic: 243.4 on 3 and 63 DF,  p-value: < 2.2e-16

plot(gasmod5sqrt,which=c(1,2))

Removing observations from analysis by subsetting the data

#We can remove particular observations
subsetgas<-gasturbine[-c(11,36,61,64),]
gasmod5rem<-lm(HEATRATE~RPM+INLET.TEMP+EXH.TEMP,data=subsetgas)
summary(gasmod5rem)


Call:
lm(formula = HEATRATE ~ RPM + INLET.TEMP + EXH.TEMP, data = subsetgas)

Residuals:
     Min       1Q   Median       3Q      Max 
-1005.32  -242.35   -89.03   208.32  1273.08 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.404e+04  6.640e+02  21.146  < 2e-16 ***
RPM          1.068e-01  1.058e-02  10.098 1.78e-14 ***
INLET.TEMP  -8.334e+00  8.352e-01  -9.979 2.79e-14 ***
EXH.TEMP     1.095e+01  2.190e+00   5.000 5.44e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 406.4 on 59 degrees of freedom
Multiple R-squared:  0.9157,    Adjusted R-squared:  0.9114 
F-statistic: 213.7 on 3 and 59 DF,  p-value: < 2.2e-16

influencePlot(gasmod5rem,fill=F)

We can cycle through a combination of variable transformations and subsetting to find a good model fit.

On Your Own

Penguins Revisited-

Consider our final model from Unit 2

\(E(bodymass)=\beta_0+\beta_1BillDep+\beta_2BillLen+\beta_3FlipperLen+\beta_4SpeciesC+\beta_5SpeciesG+\beta_6SexM+\beta_7BillDep*BillLen\)

where Species C= 1 if Chinstrap, 0 otherwise
SpeciesG = 1 if Gentoo, 0 otherwise
SexM = 1 if male, 0 otherwise

First fit the model and store it as “penmodfinal”

Checking assumptions: Check the model assumptions by fitting the model.

Is the mean zero assumption violated? How do you know?
Is the constant variance assumption violated? How do you know?
Is the normality assumptions violated? How do you know?
Is the independence assumption violated? How do you know?
If you noted any violations, attempt to correct them.

Perform the residual analysis. Identify any all outliers and/or influential observations. Include justification. Write a few sentences about your suggestions for moving forward in the analysis.