Recipe 6:Analysis of covariance
Uzma Mushtaque
October 23, 2014
Setting
Design: The purpose of this experiment is to investigate the amount of Superplasticizer used in making concrete dependent on the overall age(in days) of the concrete mixture. The analysis uses the strength (in MPa) and amount of water utilized to make the the mixture as the explanatory variable (cannot be controlled).Essentially we are trying to compare the different amounts of Superplasticizer used(kg in m^3 mixture) and their effect on the overall age of the mixture. Now strength and water used (superplasticizer are fluids that reduce moisture usage and still provide strength to th mixture) are known to be affected by the proportion of Superplasticizer used in making the concrete mixture but we want to see its effect on the age of the mixture (in days).For this anlysis we use analysis of covariance (ANCOVA) technique.
Data Summary and Exploratory data analysis
The data is a subset of a large dataset involving strength of materials. We consider only the composition and strength of concrete mixture. Independent variables are the various constituents (7 in this case)that are used to make the concrete mixture used in a variety of construction applications (buildings and bridges).
Data Type: multivariate
Abstract: Concrete is the most important material in civil engineering. The concrete compressive strength is a highly nonlinear function of age and ingredients. These ingredients include cement, blast furnace slag, fly ash, water, superplasticizer, coarse aggregate, and fine aggregate.
Source:See References
Data Characteristics(Randomization scheme):
The actual concrete compressive strength (MPa) for a given mixture under a specific age (days) was determined from laboratory. Data is in raw form (not scaled).
Summary Statistics:
Number of instances (observations): 1030 Number of Attributes: 9 Attribute breakdown: 8 quantitative input variables, and 1 quantitative output variable Missing Attribute Values: None
A summmary of the given dataset and exploratory data analysis is presented here
# Selecting the data file from the local machine
dataf<- (read.csv(file.choose(), header=T))
summary(dataf)## Cement Blast.Furnace.Slag Fly.Ash Water
## Min. :102.0 Min. : 0.0 Min. : 0.00 Min. :121.8
## 1st Qu.:192.4 1st Qu.: 0.0 1st Qu.: 0.00 1st Qu.:164.9
## Median :272.9 Median : 22.0 Median : 0.00 Median :185.0
## Mean :281.2 Mean : 73.9 Mean : 54.19 Mean :181.6
## 3rd Qu.:350.0 3rd Qu.:142.9 3rd Qu.:118.30 3rd Qu.:192.0
## Max. :540.0 Max. :359.4 Max. :200.10 Max. :247.0
## Superplasticizer Coarse.Aggregate Fine.Aggregate Age
## Min. : 0.000 Min. : 801.0 Min. :594.0 Min. : 1.00
## 1st Qu.: 0.000 1st Qu.: 932.0 1st Qu.:731.0 1st Qu.: 7.00
## Median : 6.400 Median : 968.0 Median :779.5 Median : 28.00
## Mean : 6.205 Mean : 972.9 Mean :773.6 Mean : 45.66
## 3rd Qu.:10.200 3rd Qu.:1029.4 3rd Qu.:824.0 3rd Qu.: 56.00
## Max. :32.200 Max. :1145.0 Max. :992.6 Max. :365.00
## strength
## Min. : 2.33
## 1st Qu.:23.71
## Median :34.45
## Mean :35.82
## 3rd Qu.:46.13
## Max. :82.60# Graphical representation
plot(dataf)
plot(dataf$Age~dataf$strength,pch=as.character(dataf$Superplasticizer),xlab="Strength in Mpa", ylab="Superplasticizer in kg/m^3")
plot(dataf$Age~dataf$Water,pch=as.character(dataf$Superplasticizer),xlab="Water in m^3", ylab="Superplasticizer in kg/m^3")
#par(mfrow=c(1,1))
#Create histogram
hist(dataf$Superplasticizer, xlim=c(0,50), ylab = "Superplasticizer in kg/m^3")
hist(dataf$strength, xlim=c(0,100), ylab = "Strength in MPa")
hist(dataf$Water, xlim=c(0,400), ylab = "Strength in MPa")
#Boxplots
boxplot(dataf$Age~dataf$Superplasticizer,data=dataf, xlab="Amount of Sperplasticizer in kg/m^3", ylab="Age in days")
title("Boxplot of Age variability due to Superplasticizer")
As is evident from the scatter above plot that there is no clear relationship between the amount of superplasticizer used and the age of the mixture and that is why we conduct this analysis. Also, the boxplot depicts variability of different superplasticizer quantities with that of age therefore we formulate our null hypothesis as:
There is no effect of the amount of superplasticizer used on the age of the concrete mixture.
Factors and Levels
As mentioned before, the dataset has 7 factors with multiple levels.Response variable is the age of the concrete mixture.
# Defining the factors
dataf$Superplasticizer=as.factor(dataf$Superplasticizer)Testing-Analysis of covariance (ANCOVA)
ANCOVA can be used to increase statistical power i.e. the ability to find a significant difference between groups when one exists by reducing the within-group variance.The assumptions that undelie the analysis are: normality,homogeniety of variances,homogeneity of regression slopes, linearity of regression, Independence of error terms.
datafaov=lm(Age~strength+Water+Superplasticizer,data=dataf)
anova(datafaov)## Analysis of Variance Table
##
## Response: Age
## Df Sum Sq Mean Sq F value Pr(>F)
## strength 1 444112 444112 158.1655 < 2e-16 ***
## Water 1 623166 623166 221.9337 < 2e-16 ***
## Superplasticizer 110 464044 4219 1.5024 0.00117 **
## Residuals 917 2574839 2808
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Conclusion: From the results we can reject the null hypothesis as there is a clear effect of variability in superplasticizer weight on the overall age of the concrete mixture based on the small p-value (less than 0.05) that cannot be attributed to randomization alone.
Estimation of Parameters
Parameter estimation involves calculation of contrasts. We calculate the estimates and the confidence intervals for the co-efficients in the linear model.Basically, the contrast specifies how the levels of the factors will be coded into a family of numeric dummy variables for fitting the model.
contrasts(dataf$Superplasticizer)=contr.sum
datafaov=lm(Age~strength+Water+Superplasticizer,data=dataf)
summary(datafaov)##
## Call:
## lm(formula = Age ~ strength + Water + Superplasticizer, data = dataf)
##
## Residuals:
## Min 1Q Median 3Q Max
## -122.269 -23.357 -5.936 13.389 313.347
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -289.5627 23.5746 -12.283 < 2e-16 ***
## strength 2.1991 0.1201 18.314 < 2e-16 ***
## Water 1.3681 0.1328 10.304 < 2e-16 ***
## Superplasticizer1 22.0172 5.3445 4.120 4.14e-05 ***
## Superplasticizer2 -7.3288 26.9405 -0.272 0.785656
## Superplasticizer3 -2.5652 52.8234 -0.049 0.961280
## Superplasticizer4 24.4159 52.6294 0.464 0.642814
## Superplasticizer5 24.1423 52.6292 0.459 0.646541
## Superplasticizer6 -60.0775 37.5038 -1.602 0.109521
## Superplasticizer7 -26.6360 21.8525 -1.219 0.223197
## Superplasticizer8 -13.8185 52.7090 -0.262 0.793252
## Superplasticizer9 -14.9696 30.5654 -0.490 0.624424
## Superplasticizer10 -1.1187 23.7214 -0.047 0.962396
## Superplasticizer11 -67.4538 19.3371 -3.488 0.000509 ***
## Superplasticizer12 -22.5681 52.9144 -0.427 0.669842
## Superplasticizer13 45.6366 23.8182 1.916 0.055671 .
## Superplasticizer14 24.3313 17.0482 1.427 0.153861
## Superplasticizer15 1.0877 16.5112 0.066 0.947493
## Superplasticizer16 -29.0106 26.5454 -1.093 0.274739
## Superplasticizer17 -40.8177 37.3260 -1.094 0.274442
## Superplasticizer18 5.2697 23.8697 0.221 0.825323
## Superplasticizer19 -1.3761 16.2467 -0.085 0.932517
## Superplasticizer20 51.4019 15.6793 3.278 0.001083 **
## Superplasticizer21 -5.5348 16.4460 -0.337 0.736539
## Superplasticizer22 -43.0940 30.5473 -1.411 0.158663
## Superplasticizer23 -38.4152 13.8525 -2.773 0.005664 **
## Superplasticizer24 -1.0147 20.3321 -0.050 0.960207
## Superplasticizer25 -48.0230 52.8862 -0.908 0.364094
## Superplasticizer26 -15.1272 52.7514 -0.287 0.774358
## Superplasticizer27 1.5156 21.7797 0.070 0.944535
## Superplasticizer28 9.6356 21.6880 0.444 0.656945
## Superplasticizer29 -89.1013 52.7613 -1.689 0.091605 .
## Superplasticizer30 9.4512 16.1890 0.584 0.559496
## Superplasticizer31 -15.2912 21.7170 -0.704 0.481542
## Superplasticizer32 -17.2327 12.6847 -1.359 0.174625
## Superplasticizer33 15.6126 52.6582 0.296 0.766924
## Superplasticizer34 -2.7002 52.6143 -0.051 0.959081
## Superplasticizer35 -36.6054 52.7335 -0.694 0.487758
## Superplasticizer36 33.3033 16.2089 2.055 0.040197 *
## Superplasticizer37 24.8116 21.6996 1.143 0.253165
## Superplasticizer38 27.3433 13.5572 2.017 0.043999 *
## Superplasticizer39 20.9030 21.6969 0.963 0.335596
## Superplasticizer40 -0.2388 10.6645 -0.022 0.982142
## Superplasticizer41 -13.1981 21.6993 -0.608 0.543188
## Superplasticizer42 25.7777 21.7010 1.188 0.235197
## Superplasticizer43 -8.0381 37.2764 -0.216 0.829321
## Superplasticizer44 -8.7153 52.6731 -0.165 0.868617
## Superplasticizer45 -36.8547 21.7284 -1.696 0.090196 .
## Superplasticizer46 -28.9974 21.7316 -1.334 0.182423
## Superplasticizer47 42.7881 23.7915 1.798 0.072432 .
## Superplasticizer48 -2.6184 52.7994 -0.050 0.960458
## Superplasticizer49 -29.6607 13.6127 -2.179 0.029593 *
## Superplasticizer50 -20.1428 13.6726 -1.473 0.141035
## Superplasticizer51 17.7679 26.7337 0.665 0.506457
## Superplasticizer52 -23.3726 37.3445 -0.626 0.531559
## Superplasticizer53 17.9938 23.7151 0.759 0.448199
## Superplasticizer54 -16.9631 14.9649 -1.134 0.257290
## Superplasticizer55 10.1927 21.7744 0.468 0.639821
## Superplasticizer56 13.5013 23.7065 0.570 0.569144
## Superplasticizer57 -16.8240 52.7081 -0.319 0.749654
## Superplasticizer58 31.5098 13.5782 2.321 0.020526 *
## Superplasticizer59 -11.5448 14.0783 -0.820 0.412405
## Superplasticizer60 -20.5438 20.1360 -1.020 0.307878
## Superplasticizer61 35.2176 23.8306 1.478 0.139797
## Superplasticizer62 -47.3300 21.7915 -2.172 0.030115 *
## Superplasticizer63 7.8628 16.9343 0.464 0.642532
## Superplasticizer64 -54.4327 53.0071 -1.027 0.304741
## Superplasticizer65 -25.2020 52.7253 -0.478 0.632773
## Superplasticizer66 -11.2665 52.6182 -0.214 0.830502
## Superplasticizer67 32.9863 21.7238 1.518 0.129249
## Superplasticizer68 -4.1067 17.9753 -0.228 0.819338
## Superplasticizer69 -19.2956 14.7647 -1.307 0.191584
## Superplasticizer70 -44.5140 20.2558 -2.198 0.028228 *
## Superplasticizer71 -2.7906 17.2602 -0.162 0.871597
## Superplasticizer72 13.6480 18.8627 0.724 0.469530
## Superplasticizer73 -12.7183 24.1315 -0.527 0.598291
## Superplasticizer74 -14.0784 52.6243 -0.268 0.789123
## Superplasticizer75 -20.5019 9.4886 -2.161 0.030978 *
## Superplasticizer76 26.9927 16.2668 1.659 0.097383 .
## Superplasticizer77 13.2872 23.7733 0.559 0.576358
## Superplasticizer78 24.2746 21.9983 1.103 0.270111
## Superplasticizer79 23.5277 17.9397 1.311 0.190021
## Superplasticizer80 10.9178 23.8064 0.459 0.646626
## Superplasticizer81 0.3957 52.6135 0.008 0.994001
## Superplasticizer82 -2.7550 52.7843 -0.052 0.958386
## Superplasticizer83 51.6679 23.9783 2.155 0.031439 *
## Superplasticizer84 20.5216 37.3202 0.550 0.582536
## Superplasticizer85 -16.4656 52.6263 -0.313 0.754445
## Superplasticizer86 35.4620 21.7679 1.629 0.103635
## Superplasticizer87 9.8014 26.4752 0.370 0.711311
## Superplasticizer88 22.3592 52.7476 0.424 0.671745
## Superplasticizer89 6.9728 26.5436 0.263 0.792845
## Superplasticizer90 27.8106 23.9930 1.159 0.246712
## Superplasticizer91 5.4829 17.9651 0.305 0.760285
## Superplasticizer92 26.3621 26.5429 0.993 0.320880
## Superplasticizer93 -38.4971 52.6548 -0.731 0.464891
## Superplasticizer94 59.8992 52.7031 1.137 0.256027
## Superplasticizer95 10.7604 23.8710 0.451 0.652260
## Superplasticizer96 60.7082 37.3980 1.623 0.104869
## Superplasticizer97 61.7688 52.7047 1.172 0.241510
## Superplasticizer98 -3.2463 14.3155 -0.227 0.820654
## Superplasticizer99 26.0290 37.3067 0.698 0.485540
## Superplasticizer100 37.6368 37.3261 1.008 0.313564
## Superplasticizer101 8.4663 52.6157 0.161 0.872201
## Superplasticizer102 -2.1810 23.9694 -0.091 0.927520
## Superplasticizer103 -1.2975 52.6184 -0.025 0.980333
## Superplasticizer104 -1.5711 52.6188 -0.030 0.976187
## Superplasticizer105 -20.7603 52.6887 -0.394 0.693660
## Superplasticizer106 -33.2931 52.6558 -0.632 0.527362
## Superplasticizer107 -15.1528 22.0868 -0.686 0.492851
## Superplasticizer108 3.9435 52.6144 0.075 0.940270
## Superplasticizer109 41.6002 24.5514 1.694 0.090526 .
## Superplasticizer110 17.0664 24.1327 0.707 0.479627
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 52.99 on 917 degrees of freedom
## Multiple R-squared: 0.3729, Adjusted R-squared: 0.2963
## F-statistic: 4.869 on 112 and 917 DF, p-value: < 2.2e-16confint(datafaov)## 2.5 % 97.5 %
## (Intercept) -335.8291601 -243.296224
## strength 1.9634786 2.434811
## Water 1.1075152 1.628693
## Superplasticizer1 11.5281957 32.506104
## Superplasticizer2 -60.2009998 45.543454
## Superplasticizer3 -106.2339111 101.103597
## Superplasticizer4 -78.8720819 127.703881
## Superplasticizer5 -79.1454597 127.430018
## Superplasticizer6 -133.6806905 13.525659
## Superplasticizer7 -69.5227958 16.250794
## Superplasticizer8 -117.2628746 89.625783
## Superplasticizer9 -74.9557779 45.016670
## Superplasticizer10 -47.6732964 45.435920
## Superplasticizer11 -105.4038795 -29.503766
## Superplasticizer12 -126.4154259 81.279214
## Superplasticizer13 -1.1078737 92.381126
## Superplasticizer14 -9.1266949 57.789307
## Superplasticizer15 -31.3165224 33.491835
## Superplasticizer16 -81.1073943 23.086162
## Superplasticizer17 -114.0720496 32.436677
## Superplasticizer18 -41.5759877 52.115301
## Superplasticizer19 -33.2610493 30.508829
## Superplasticizer20 20.6304740 82.173316
## Superplasticizer21 -37.8108381 26.741334
## Superplasticizer22 -103.0447160 16.856691
## Superplasticizer23 -65.6015863 -11.228848
## Superplasticizer24 -40.9175706 38.888103
## Superplasticizer25 -151.8150829 55.769145
## Superplasticizer26 -118.6546029 88.400257
## Superplasticizer27 -41.2281654 44.259461
## Superplasticizer28 -32.9283626 52.199474
## Superplasticizer29 -192.6480937 14.445553
## Superplasticizer30 -22.3205753 41.222932
## Superplasticizer31 -57.9120380 27.329647
## Superplasticizer32 -42.1270415 7.661675
## Superplasticizer33 -87.7319946 118.957099
## Superplasticizer34 -105.9585642 100.558090
## Superplasticizer35 -140.0977229 66.886901
## Superplasticizer36 1.4925020 65.114176
## Superplasticizer37 -17.7749669 67.398214
## Superplasticizer38 0.7365212 53.950027
## Superplasticizer39 -21.6782870 63.484240
## Superplasticizer40 -21.1684210 20.690877
## Superplasticizer41 -55.7841301 29.387943
## Superplasticizer42 -16.8116490 68.367054
## Superplasticizer43 -81.1949524 65.118814
## Superplasticizer44 -112.0891222 94.658423
## Superplasticizer45 -79.4977715 5.788380
## Superplasticizer46 -71.6468636 13.652058
## Superplasticizer47 -3.9039533 89.480240
## Superplasticizer48 -106.2402279 101.003336
## Superplasticizer49 -56.3764050 -2.944991
## Superplasticizer50 -46.9760503 6.690472
## Superplasticizer51 -34.6984279 70.234223
## Superplasticizer52 -96.6632542 49.917955
## Superplasticizer53 -28.5484721 64.536046
## Superplasticizer54 -46.3324989 12.406328
## Superplasticizer55 -32.5408120 52.926245
## Superplasticizer56 -33.0239830 60.026506
## Superplasticizer57 -120.2665419 86.618606
## Superplasticizer58 4.8619420 58.157706
## Superplasticizer59 -39.1743313 16.084656
## Superplasticizer60 -60.0617319 18.974164
## Superplasticizer61 -11.5513376 81.986438
## Superplasticizer62 -90.0970140 -4.562996
## Superplasticizer63 -25.3715583 41.097258
## Superplasticizer64 -158.4620266 49.596622
## Superplasticizer65 -128.6782340 78.274179
## Superplasticizer66 -114.5326992 91.999618
## Superplasticizer67 -9.6478979 75.620468
## Superplasticizer68 -39.3842393 31.170875
## Superplasticizer69 -48.2721977 9.680976
## Superplasticizer70 -84.2671567 -4.760881
## Superplasticizer71 -36.6645878 31.083478
## Superplasticizer72 -23.3711061 50.667153
## Superplasticizer73 -60.0777426 34.641054
## Superplasticizer74 -117.3565734 89.199680
## Superplasticizer75 -39.1238175 -1.880044
## Superplasticizer76 -4.9318466 58.917207
## Superplasticizer77 -33.3691832 59.943561
## Superplasticizer78 -18.8983177 67.447430
## Superplasticizer79 -11.6799111 58.735376
## Superplasticizer80 -35.8036538 57.639221
## Superplasticizer81 -102.8610797 103.652470
## Superplasticizer82 -106.3470056 100.837021
## Superplasticizer83 4.6091440 98.726632
## Superplasticizer84 -52.7213076 93.764593
## Superplasticizer85 -119.7476313 86.816504
## Superplasticizer86 -7.2585460 78.182643
## Superplasticizer87 -42.1575730 61.760377
## Superplasticizer88 -81.1608265 125.879309
## Superplasticizer89 -45.1203930 59.066073
## Superplasticizer90 -19.2770094 74.898287
## Superplasticizer91 -29.7745408 40.740355
## Superplasticizer92 -25.7297871 78.454007
## Superplasticizer93 -141.8349411 64.840769
## Superplasticizer94 -43.5334264 163.331900
## Superplasticizer95 -36.0877877 57.608567
## Superplasticizer96 -12.6873620 134.103750
## Superplasticizer97 -41.6669980 165.204560
## Superplasticizer98 -31.3412523 24.848634
## Superplasticizer99 -47.1874635 99.245370
## Superplasticizer100 -35.6176531 110.891349
## Superplasticizer101 -94.7947610 111.727419
## Superplasticizer102 -49.2223137 44.860339
## Superplasticizer103 -104.5638648 101.968939
## Superplasticizer104 -104.8382481 101.696080
## Superplasticizer105 -124.1647470 82.644225
## Superplasticizer106 -136.6329105 70.046684
## Superplasticizer107 -58.4994793 28.193806
## Superplasticizer108 -99.3151587 107.202153
## Superplasticizer109 -6.5832477 89.783578
## Superplasticizer110 -30.2953019 64.428173So from our results we can see that the value of µ, beta, alpha1, alpha2 are -289.5,2.2,1.3,22.7.Essentially if contrasts used here are changed, the model will be the same in terms of the underlying probability distributions allowed. Only the parameters will be different. The fitted values will stay the same as well. So once we have the value of the parameters for one choice of contrasts, it is easy to estimate the value of the parameters for another choice of contrasts.
Diagnostics/Model Adequacy Checking
Model adequacy checking is done to test the validity of ANCOVA given the numerous assumptions it is based on.
The shapiro test evaluates the Null hypothesis such that “the samples come from a Normal distribution” against the alternative hypothesis “the samples do not come from a Normal distribution”.
The Q-Q Norm plots also test the normality of the sample. The scatter plots of the residuals and the fitted model will check the distribution of the model over the entire dynamic range.
We also fit a model with different slopes for each factor level.
# Shapiro Test
shapiro.test(dataf$Age)##
## Shapiro-Wilk normality test
##
## data: dataf$Age
## W = 0.5907, p-value < 2.2e-16#Plots for checking normality
qqnorm(residuals(datafaov))
qqline(residuals(datafaov))
plot(fitted(datafaov),residuals(datafaov))
# Fit a model with different slopes for each factor level
aovdatafnew=lm(Age~Superplasticizer*Water*strength,data=dataf)
summary(aovdatafnew)##
## Call:
## lm(formula = Age ~ Superplasticizer * Water * strength, data = dataf)
##
## Residuals:
## Min 1Q Median 3Q Max
## -172.93 -13.99 0.00 8.58 320.43
##
## Coefficients: (167 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.486e+02 2.911e+05 -0.002 0.999
## Superplasticizer1 6.716e+02 2.911e+05 0.002 0.998
## Superplasticizer2 -5.757e+01 1.519e+05 0.000 1.000
## Superplasticizer3 1.453e+03 3.134e+06 0.000 1.000
## Superplasticizer4 -1.744e+01 2.736e+04 -0.001 0.999
## Superplasticizer5 -1.790e+01 2.744e+04 -0.001 0.999
## Superplasticizer6 1.075e+02 5.762e+04 0.002 0.999
## Superplasticizer7 4.766e+02 2.911e+05 0.002 0.999
## Superplasticizer8 -8.155e+01 3.096e+04 -0.003 0.998
## Superplasticizer9 4.766e+02 8.897e+04 0.005 0.996
## Superplasticizer10 -3.040e+01 9.590e+04 0.000 1.000
## Superplasticizer11 2.187e+02 2.911e+05 0.001 0.999
## Superplasticizer12 -1.045e+02 1.910e+04 -0.005 0.996
## Superplasticizer13 -2.519e+01 4.656e+04 -0.001 1.000
## Superplasticizer14 3.912e+02 2.911e+05 0.001 0.999
## Superplasticizer15 2.771e+03 2.911e+05 0.010 0.992
## Superplasticizer16 4.764e+02 4.179e+06 0.000 1.000
## Superplasticizer17 -9.783e+02 4.126e+05 -0.002 0.998
## Superplasticizer18 -2.048e+01 1.214e+05 0.000 1.000
## Superplasticizer19 9.517e+01 2.911e+05 0.000 1.000
## Superplasticizer20 4.022e+02 2.911e+05 0.001 0.999
## Superplasticizer21 -1.282e+03 2.911e+05 -0.004 0.996
## Superplasticizer22 4.766e+02 2.605e+05 0.002 0.999
## Superplasticizer23 4.766e+02 2.911e+05 0.002 0.999
## Superplasticizer24 2.034e+03 2.912e+05 0.007 0.994
## Superplasticizer25 -1.395e+02 1.668e+04 -0.008 0.993
## Superplasticizer26 -8.328e+01 3.978e+04 -0.002 0.998
## Superplasticizer27 -7.496e+02 1.829e+05 -0.004 0.997
## Superplasticizer28 3.432e+02 6.475e+04 0.005 0.996
## Superplasticizer29 -2.149e+02 8.829e+04 -0.002 0.998
## Superplasticizer30 -2.347e+02 2.911e+05 -0.001 0.999
## Superplasticizer31 3.124e+03 3.678e+05 0.008 0.993
## Superplasticizer32 3.670e+02 2.911e+05 0.001 0.999
## Superplasticizer33 -3.137e+01 4.107e+04 -0.001 0.999
## Superplasticizer34 -6.563e+01 7.490e+03 -0.009 0.993
## Superplasticizer35 -1.211e+02 8.124e+03 -0.015 0.988
## Superplasticizer36 -7.470e+02 2.911e+05 -0.003 0.998
## Superplasticizer37 -1.880e+03 2.888e+05 -0.007 0.995
## Superplasticizer38 1.016e+02 2.911e+05 0.000 1.000
## Superplasticizer39 4.347e+01 1.637e+05 0.000 1.000
## Superplasticizer40 4.766e+02 2.911e+05 0.002 0.999
## Superplasticizer41 7.961e+02 3.415e+05 0.002 0.998
## Superplasticizer42 -3.118e+02 1.583e+05 -0.002 0.998
## Superplasticizer43 1.587e+02 5.199e+04 0.003 0.998
## Superplasticizer44 -7.322e+01 2.711e+04 -0.003 0.998
## Superplasticizer45 -5.094e+02 2.911e+05 -0.002 0.999
## Superplasticizer46 1.785e+02 7.210e+04 0.002 0.998
## Superplasticizer47 3.927e+01 5.187e+04 0.001 0.999
## Superplasticizer48 -6.097e+01 6.667e+04 -0.001 0.999
## Superplasticizer49 1.033e+03 2.911e+05 0.004 0.997
## Superplasticizer50 4.766e+02 2.911e+05 0.002 0.999
## Superplasticizer51 3.616e+01 1.246e+05 0.000 1.000
## Superplasticizer52 -1.431e+03 1.381e+06 -0.001 0.999
## Superplasticizer53 -2.174e+01 7.064e+04 0.000 1.000
## Superplasticizer54 4.541e+01 2.911e+05 0.000 1.000
## Superplasticizer55 6.062e+02 2.297e+05 0.003 0.998
## Superplasticizer56 9.397e+00 7.963e+04 0.000 1.000
## Superplasticizer57 -8.681e+01 2.710e+04 -0.003 0.997
## Superplasticizer58 2.289e+03 2.911e+05 0.008 0.994
## Superplasticizer59 4.766e+02 2.911e+05 0.002 0.999
## Superplasticizer60 -1.701e+04 2.941e+05 -0.058 0.954
## Superplasticizer61 3.891e+01 4.232e+04 0.001 0.999
## Superplasticizer62 -8.236e+03 2.921e+05 -0.028 0.978
## Superplasticizer63 -3.265e+02 2.911e+05 -0.001 0.999
## Superplasticizer64 -1.497e+02 2.631e+04 -0.006 0.995
## Superplasticizer65 -1.012e+02 2.034e+04 -0.005 0.996
## Superplasticizer66 -7.912e+01 8.731e+03 -0.009 0.993
## Superplasticizer67 5.460e+02 9.935e+04 0.005 0.996
## Superplasticizer68 3.582e+02 1.474e+05 0.002 0.998
## Superplasticizer69 4.766e+02 2.911e+05 0.002 0.999
## Superplasticizer70 6.489e+03 2.911e+05 0.022 0.982
## Superplasticizer71 6.389e+02 2.911e+05 0.002 0.998
## Superplasticizer72 -2.234e+03 2.911e+05 -0.008 0.994
## Superplasticizer73 1.566e+01 2.680e+04 0.001 1.000
## Superplasticizer74 -8.379e+01 8.582e+03 -0.010 0.992
## Superplasticizer75 -7.251e+02 2.911e+05 -0.002 0.998
## Superplasticizer76 1.674e+02 2.911e+05 0.001 1.000
## Superplasticizer77 1.431e+01 5.464e+04 0.000 1.000
## Superplasticizer78 3.432e+02 4.578e+04 0.007 0.994
## Superplasticizer79 5.036e+01 2.911e+05 0.000 1.000
## Superplasticizer80 1.294e+01 4.890e+04 0.000 1.000
## Superplasticizer81 -6.015e+01 8.335e+03 -0.007 0.994
## Superplasticizer82 -6.886e+01 1.185e+04 -0.006 0.995
## Superplasticizer83 3.326e+01 2.413e+04 0.001 0.999
## Superplasticizer84 -1.908e+03 1.455e+06 -0.001 0.999
## Superplasticizer85 -8.790e+01 7.776e+03 -0.011 0.991
## Superplasticizer86 9.261e+02 7.116e+04 0.013 0.990
## Superplasticizer87 4.766e+02 2.928e+05 0.002 0.999
## Superplasticizer88 -2.456e+01 1.375e+04 -0.002 0.999
## Superplasticizer89 5.534e+01 4.698e+04 0.001 0.999
## Superplasticizer90 5.751e+01 2.454e+04 0.002 0.998
## Superplasticizer91 -8.442e+02 2.914e+05 -0.003 0.998
## Superplasticizer92 4.766e+02 1.129e+05 0.004 0.997
## Superplasticizer93 -1.260e+02 1.971e+04 -0.006 0.995
## Superplasticizer94 4.456e+01 4.646e+04 0.001 0.999
## Superplasticizer95 1.970e+01 4.042e+04 0.000 1.000
## Superplasticizer96 1.012e+02 2.554e+05 0.000 1.000
## Superplasticizer97 4.670e+01 3.098e+04 0.002 0.999
## Superplasticizer98 1.523e+01 3.957e+04 0.000 1.000
## Superplasticizer99 2.693e+03 7.918e+05 0.003 0.997
## Superplasticizer100 1.699e+03 4.012e+05 0.004 0.997
## Superplasticizer101 -4.604e+01 1.191e+04 -0.004 0.997
## Superplasticizer102 1.059e+01 3.515e+04 0.000 1.000
## Superplasticizer103 -6.174e+01 1.544e+04 -0.004 0.997
## Superplasticizer104 -6.220e+01 1.543e+04 -0.004 0.997
## Superplasticizer105 -9.886e+01 1.149e+04 -0.009 0.993
## Superplasticizer106 -1.198e+02 2.277e+04 -0.005 0.996
## Superplasticizer107 3.420e+01 2.372e+04 0.001 0.999
## Superplasticizer108 -5.398e+01 9.627e+03 -0.006 0.996
## Superplasticizer109 8.689e+01 2.029e+04 0.004 0.997
## Superplasticizer110 -4.120e+01 1.558e+04 -0.003 0.998
## Water 2.278e+00 2.149e+03 0.001 0.999
## strength 3.879e+00 9.045e+03 0.000 1.000
## Superplasticizer1:Water -3.621e+00 2.149e+03 -0.002 0.999
## Superplasticizer2:Water NA NA NA NA
## Superplasticizer3:Water NA NA NA NA
## Superplasticizer4:Water NA NA NA NA
## Superplasticizer5:Water NA NA NA NA
## Superplasticizer6:Water NA NA NA NA
## Superplasticizer7:Water -2.278e+00 2.149e+03 -0.001 0.999
## Superplasticizer8:Water NA NA NA NA
## Superplasticizer9:Water -2.278e+00 6.205e+01 -0.037 0.971
## Superplasticizer10:Water NA NA NA NA
## Superplasticizer11:Water -1.351e+00 2.149e+03 -0.001 0.999
## Superplasticizer12:Water NA NA NA NA
## Superplasticizer13:Water NA NA NA NA
## Superplasticizer14:Water -2.243e+00 2.149e+03 -0.001 0.999
## Superplasticizer15:Water -1.434e+01 2.149e+03 -0.007 0.995
## Superplasticizer16:Water -2.278e+00 2.202e+04 0.000 1.000
## Superplasticizer17:Water 4.905e+00 2.226e+03 0.002 0.998
## Superplasticizer18:Water NA NA NA NA
## Superplasticizer19:Water -4.748e-01 2.149e+03 0.000 1.000
## Superplasticizer20:Water -2.199e+00 2.149e+03 -0.001 0.999
## Superplasticizer21:Water 6.501e+00 2.149e+03 0.003 0.998
## Superplasticizer22:Water -2.278e+00 1.176e+03 -0.002 0.998
## Superplasticizer23:Water -2.278e+00 2.149e+03 -0.001 0.999
## Superplasticizer24:Water -1.069e+01 2.150e+03 -0.005 0.996
## Superplasticizer25:Water NA NA NA NA
## Superplasticizer26:Water NA NA NA NA
## Superplasticizer27:Water 3.813e+00 3.643e+02 0.010 0.992
## Superplasticizer28:Water -2.032e+00 8.211e+00 -0.247 0.805
## Superplasticizer29:Water NA NA NA NA
## Superplasticizer30:Water 1.335e+00 2.149e+03 0.001 1.000
## Superplasticizer31:Water -1.726e+01 1.476e+03 -0.012 0.991
## Superplasticizer32:Water -1.795e+00 2.149e+03 -0.001 0.999
## Superplasticizer33:Water NA NA NA NA
## Superplasticizer34:Water NA NA NA NA
## Superplasticizer35:Water NA NA NA NA
## Superplasticizer36:Water 4.584e+00 2.149e+03 0.002 0.998
## Superplasticizer37:Water 1.117e+01 1.269e+03 0.009 0.993
## Superplasticizer38:Water -4.377e-01 2.149e+03 0.000 1.000
## Superplasticizer39:Water -6.568e-02 1.372e+03 0.000 1.000
## Superplasticizer40:Water -2.278e+00 2.149e+03 -0.001 0.999
## Superplasticizer41:Water -4.548e+00 1.463e+03 -0.003 0.998
## Superplasticizer42:Water 2.044e+00 4.899e+02 0.004 0.997
## Superplasticizer43:Water -1.306e+00 3.414e+02 -0.004 0.997
## Superplasticizer44:Water NA NA NA NA
## Superplasticizer45:Water 2.863e+00 2.149e+03 0.001 0.999
## Superplasticizer46:Water -1.101e+00 8.240e+01 -0.013 0.989
## Superplasticizer47:Water NA NA NA NA
## Superplasticizer48:Water NA NA NA NA
## Superplasticizer49:Water -5.794e+00 2.149e+03 -0.003 0.998
## Superplasticizer50:Water -2.278e+00 2.149e+03 -0.001 0.999
## Superplasticizer51:Water NA NA NA NA
## Superplasticizer52:Water 6.986e+00 7.278e+03 0.001 0.999
## Superplasticizer53:Water NA NA NA NA
## Superplasticizer54:Water -2.780e-01 2.149e+03 0.000 1.000
## Superplasticizer55:Water -3.559e+00 1.197e+03 -0.003 0.998
## Superplasticizer56:Water NA NA NA NA
## Superplasticizer57:Water NA NA NA NA
## Superplasticizer58:Water -1.381e+01 2.149e+03 -0.006 0.995
## Superplasticizer59:Water -2.278e+00 2.149e+03 -0.001 0.999
## Superplasticizer60:Water 9.987e+01 2.163e+03 0.046 0.963
## Superplasticizer61:Water NA NA NA NA
## Superplasticizer62:Water 4.481e+01 2.153e+03 0.021 0.983
## Superplasticizer63:Water 2.595e+00 2.149e+03 0.001 0.999
## Superplasticizer64:Water NA NA NA NA
## Superplasticizer65:Water NA NA NA NA
## Superplasticizer66:Water NA NA NA NA
## Superplasticizer67:Water -3.074e+00 3.151e+02 -0.010 0.992
## Superplasticizer68:Water -1.878e+00 6.965e+02 -0.003 0.998
## Superplasticizer69:Water -2.278e+00 2.149e+03 -0.001 0.999
## Superplasticizer70:Water -3.527e+01 2.149e+03 -0.016 0.987
## Superplasticizer71:Water -3.862e+00 2.149e+03 -0.002 0.999
## Superplasticizer72:Water 1.432e+01 2.149e+03 0.007 0.995
## Superplasticizer73:Water NA NA NA NA
## Superplasticizer74:Water NA NA NA NA
## Superplasticizer75:Water 4.668e+00 2.149e+03 0.002 0.998
## Superplasticizer76:Water -7.625e-01 2.149e+03 0.000 1.000
## Superplasticizer77:Water NA NA NA NA
## Superplasticizer78:Water -1.946e+00 2.386e+02 -0.008 0.993
## Superplasticizer79:Water 1.494e-01 2.149e+03 0.000 1.000
## Superplasticizer80:Water NA NA NA NA
## Superplasticizer81:Water NA NA NA NA
## Superplasticizer82:Water NA NA NA NA
## Superplasticizer83:Water NA NA NA NA
## Superplasticizer84:Water 1.056e+01 8.363e+03 0.001 0.999
## Superplasticizer85:Water NA NA NA NA
## Superplasticizer86:Water -5.624e+00 1.395e+02 -0.040 0.968
## Superplasticizer87:Water -2.278e+00 2.156e+03 -0.001 0.999
## Superplasticizer88:Water NA NA NA NA
## Superplasticizer89:Water NA NA NA NA
## Superplasticizer90:Water NA NA NA NA
## Superplasticizer91:Water 5.718e+00 2.151e+03 0.003 0.998
## Superplasticizer92:Water -2.278e+00 1.119e+03 -0.002 0.998
## Superplasticizer93:Water NA NA NA NA
## Superplasticizer94:Water NA NA NA NA
## Superplasticizer95:Water NA NA NA NA
## Superplasticizer96:Water -3.435e-01 1.806e+03 0.000 1.000
## Superplasticizer97:Water NA NA NA NA
## Superplasticizer98:Water NA NA NA NA
## Superplasticizer99:Water -1.665e+01 4.789e+03 -0.003 0.997
## Superplasticizer100:Water -1.040e+01 2.332e+03 -0.004 0.996
## Superplasticizer101:Water NA NA NA NA
## Superplasticizer102:Water NA NA NA NA
## Superplasticizer103:Water NA NA NA NA
## Superplasticizer104:Water NA NA NA NA
## Superplasticizer105:Water NA NA NA NA
## Superplasticizer106:Water NA NA NA NA
## Superplasticizer107:Water NA NA NA NA
## Superplasticizer108:Water NA NA NA NA
## Superplasticizer109:Water NA NA NA NA
## Superplasticizer110:Water NA NA NA NA
## Superplasticizer1:strength -1.568e+01 9.045e+03 -0.002 0.999
## Superplasticizer2:strength -4.400e-01 4.095e+03 0.000 1.000
## Superplasticizer3:strength -2.660e+01 5.503e+04 0.000 1.000
## Superplasticizer4:strength NA NA NA NA
## Superplasticizer5:strength NA NA NA NA
## Superplasticizer6:strength -3.879e+00 1.288e+03 -0.003 0.998
## Superplasticizer7:strength -3.879e+00 9.045e+03 0.000 1.000
## Superplasticizer8:strength NA NA NA NA
## Superplasticizer9:strength -3.879e+00 2.558e+03 -0.002 0.999
## Superplasticizer10:strength -1.043e+00 2.430e+03 0.000 1.000
## Superplasticizer11:strength -5.529e-01 9.045e+03 0.000 1.000
## Superplasticizer12:strength NA NA NA NA
## Superplasticizer13:strength 1.072e+00 9.568e+02 0.001 0.999
## Superplasticizer14:strength 4.628e+00 9.045e+03 0.001 1.000
## Superplasticizer15:strength -1.166e+02 9.045e+03 -0.013 0.990
## Superplasticizer16:strength -3.874e+00 1.248e+05 0.000 1.000
## Superplasticizer17:strength NA NA NA NA
## Superplasticizer18:strength -1.319e+00 3.189e+03 0.000 1.000
## Superplasticizer19:strength 5.365e+00 9.045e+03 0.001 1.000
## Superplasticizer20:strength 1.119e+00 9.045e+03 0.000 1.000
## Superplasticizer21:strength 4.986e+01 9.046e+03 0.006 0.996
## Superplasticizer22:strength -3.879e+00 1.052e+03 -0.004 0.997
## Superplasticizer23:strength -3.879e+00 9.045e+03 0.000 1.000
## Superplasticizer24:strength -7.597e+01 9.048e+03 -0.008 0.993
## Superplasticizer25:strength NA NA NA NA
## Superplasticizer26:strength NA NA NA NA
## Superplasticizer27:strength -8.968e-01 2.979e+03 0.000 1.000
## Superplasticizer28:strength -1.307e+00 1.467e+03 -0.001 0.999
## Superplasticizer29:strength NA NA NA NA
## Superplasticizer30:strength 2.205e+01 9.045e+03 0.002 0.998
## Superplasticizer31:strength -1.698e+00 2.545e+03 -0.001 0.999
## Superplasticizer32:strength -4.637e-01 9.045e+03 0.000 1.000
## Superplasticizer33:strength NA NA NA NA
## Superplasticizer34:strength NA NA NA NA
## Superplasticizer35:strength NA NA NA NA
## Superplasticizer36:strength 5.019e+01 9.045e+03 0.006 0.996
## Superplasticizer37:strength -7.915e-01 1.773e+03 0.000 1.000
## Superplasticizer38:strength 9.708e+00 9.045e+03 0.001 0.999
## Superplasticizer39:strength -1.819e+00 1.416e+03 -0.001 0.999
## Superplasticizer40:strength -3.879e+00 9.045e+03 0.000 1.000
## Superplasticizer41:strength -1.709e+00 2.124e+03 -0.001 0.999
## Superplasticizer42:strength -1.749e+00 1.811e+03 -0.001 0.999
## Superplasticizer43:strength NA NA NA NA
## Superplasticizer44:strength NA NA NA NA
## Superplasticizer45:strength 1.454e+01 9.045e+03 0.002 0.999
## Superplasticizer46:strength -1.968e+00 2.150e+03 -0.001 0.999
## Superplasticizer47:strength -1.121e+00 1.116e+03 -0.001 0.999
## Superplasticizer48:strength NA NA NA NA
## Superplasticizer49:strength -1.227e+01 9.045e+03 -0.001 0.999
## Superplasticizer50:strength -3.879e+00 9.045e+03 0.000 1.000
## Superplasticizer51:strength -3.879e+00 3.285e+03 -0.001 0.999
## Superplasticizer52:strength NA NA NA NA
## Superplasticizer53:strength -4.499e-01 1.678e+03 0.000 1.000
## Superplasticizer54:strength -5.381e+00 9.045e+03 -0.001 1.000
## Superplasticizer55:strength -2.014e+00 8.802e+02 -0.002 0.998
## Superplasticizer56:strength -1.508e+00 1.945e+03 -0.001 0.999
## Superplasticizer57:strength NA NA NA NA
## Superplasticizer58:strength -8.089e+01 9.045e+03 -0.009 0.993
## Superplasticizer59:strength -3.879e+00 9.045e+03 0.000 1.000
## Superplasticizer60:strength 1.036e+03 9.394e+03 0.110 0.912
## Superplasticizer61:strength -1.297e+00 8.292e+02 -0.002 0.999
## Superplasticizer62:strength 1.605e+02 9.056e+03 0.018 0.986
## Superplasticizer63:strength 2.223e+01 9.045e+03 0.002 0.998
## Superplasticizer64:strength NA NA NA NA
## Superplasticizer65:strength NA NA NA NA
## Superplasticizer66:strength NA NA NA NA
## Superplasticizer67:strength -1.576e+00 1.052e+03 -0.001 0.999
## Superplasticizer68:strength -2.784e+00 7.782e+02 -0.004 0.997
## Superplasticizer69:strength -3.879e+00 9.045e+03 0.000 1.000
## Superplasticizer70:strength -2.077e+02 9.047e+03 -0.023 0.982
## Superplasticizer71:strength -7.429e+00 9.045e+03 -0.001 0.999
## Superplasticizer72:strength 7.655e+01 9.045e+03 0.008 0.993
## Superplasticizer73:strength -1.736e+00 3.572e+02 -0.005 0.996
## Superplasticizer74:strength NA NA NA NA
## Superplasticizer75:strength 2.303e+01 9.045e+03 0.003 0.998
## Superplasticizer76:strength -5.666e-01 9.045e+03 0.000 1.000
## Superplasticizer77:strength -1.411e+00 1.199e+03 -0.001 0.999
## Superplasticizer78:strength -1.857e+00 6.389e+01 -0.029 0.977
## Superplasticizer79:strength 6.293e+00 9.045e+03 0.001 0.999
## Superplasticizer80:strength -1.387e+00 1.027e+03 -0.001 0.999
## Superplasticizer81:strength NA NA NA NA
## Superplasticizer82:strength NA NA NA NA
## Superplasticizer83:strength -4.302e-01 2.743e+02 -0.002 0.999
## Superplasticizer84:strength NA NA NA NA
## Superplasticizer85:strength NA NA NA NA
## Superplasticizer86:strength -1.153e+00 1.040e+03 -0.001 0.999
## Superplasticizer87:strength -3.879e+00 9.088e+03 0.000 1.000
## Superplasticizer88:strength NA NA NA NA
## Superplasticizer89:strength -2.457e+00 9.695e+02 -0.003 0.998
## Superplasticizer90:strength -1.763e+00 2.871e+02 -0.006 0.995
## Superplasticizer91:strength 2.673e+01 9.049e+03 0.003 0.998
## Superplasticizer92:strength -3.879e+00 2.150e+03 -0.002 0.999
## Superplasticizer93:strength NA NA NA NA
## Superplasticizer94:strength NA NA NA NA
## Superplasticizer95:strength -1.471e+00 7.718e+02 -0.002 0.998
## Superplasticizer96:strength NA NA NA NA
## Superplasticizer97:strength NA NA NA NA
## Superplasticizer98:strength -1.664e+00 7.463e+02 -0.002 0.998
## Superplasticizer99:strength NA NA NA NA
## Superplasticizer100:strength NA NA NA NA
## Superplasticizer101:strength NA NA NA NA
## Superplasticizer102:strength -1.527e+00 6.123e+02 -0.002 0.998
## Superplasticizer103:strength NA NA NA NA
## Superplasticizer104:strength NA NA NA NA
## Superplasticizer105:strength NA NA NA NA
## Superplasticizer106:strength NA NA NA NA
## Superplasticizer107:strength -2.109e+00 2.615e+02 -0.008 0.994
## Superplasticizer108:strength NA NA NA NA
## Superplasticizer109:strength -1.709e+00 9.695e+02 -0.002 0.999
## Superplasticizer110:strength NA NA NA NA
## Water:strength -4.003e-07 6.378e+01 0.000 1.000
## Superplasticizer1:Water:strength 7.783e-02 6.378e+01 0.001 0.999
## Superplasticizer2:Water:strength NA NA NA NA
## Superplasticizer3:Water:strength NA NA NA NA
## Superplasticizer4:Water:strength NA NA NA NA
## Superplasticizer5:Water:strength NA NA NA NA
## Superplasticizer6:Water:strength NA NA NA NA
## Superplasticizer7:Water:strength 4.004e-07 6.379e+01 0.000 1.000
## Superplasticizer8:Water:strength NA NA NA NA
## Superplasticizer9:Water:strength NA NA NA NA
## Superplasticizer10:Water:strength NA NA NA NA
## Superplasticizer11:Water:strength -9.430e-03 6.379e+01 0.000 1.000
## Superplasticizer12:Water:strength NA NA NA NA
## Superplasticizer13:Water:strength NA NA NA NA
## Superplasticizer14:Water:strength -2.662e-02 6.379e+01 0.000 1.000
## Superplasticizer15:Water:strength 5.923e-01 6.379e+01 0.009 0.993
## Superplasticizer16:Water:strength -2.281e-05 6.586e+02 0.000 1.000
## Superplasticizer17:Water:strength NA NA NA NA
## Superplasticizer18:Water:strength NA NA NA NA
## Superplasticizer19:Water:strength -4.115e-02 6.379e+01 -0.001 0.999
## Superplasticizer20:Water:strength -1.432e-02 6.378e+01 0.000 1.000
## Superplasticizer21:Water:strength -2.653e-01 6.379e+01 -0.004 0.997
## Superplasticizer22:Water:strength NA NA NA NA
## Superplasticizer23:Water:strength 4.003e-07 6.378e+01 0.000 1.000
## Superplasticizer24:Water:strength 3.881e-01 6.380e+01 0.006 0.995
## Superplasticizer25:Water:strength NA NA NA NA
## Superplasticizer26:Water:strength NA NA NA NA
## Superplasticizer27:Water:strength NA NA NA NA
## Superplasticizer28:Water:strength NA NA NA NA
## Superplasticizer29:Water:strength NA NA NA NA
## Superplasticizer30:Water:strength -1.322e-01 6.379e+01 -0.002 0.998
## Superplasticizer31:Water:strength NA NA NA NA
## Superplasticizer32:Water:strength -1.492e-02 6.378e+01 0.000 1.000
## Superplasticizer33:Water:strength NA NA NA NA
## Superplasticizer34:Water:strength NA NA NA NA
## Superplasticizer35:Water:strength NA NA NA NA
## Superplasticizer36:Water:strength -3.047e-01 6.379e+01 -0.005 0.996
## Superplasticizer37:Water:strength NA NA NA NA
## Superplasticizer38:Water:strength -6.728e-02 6.378e+01 -0.001 0.999
## Superplasticizer39:Water:strength NA NA NA NA
## Superplasticizer40:Water:strength 4.003e-07 6.378e+01 0.000 1.000
## Superplasticizer41:Water:strength NA NA NA NA
## Superplasticizer42:Water:strength NA NA NA NA
## Superplasticizer43:Water:strength NA NA NA NA
## Superplasticizer44:Water:strength NA NA NA NA
## Superplasticizer45:Water:strength -9.600e-02 6.379e+01 -0.002 0.999
## Superplasticizer46:Water:strength NA NA NA NA
## Superplasticizer47:Water:strength NA NA NA NA
## Superplasticizer48:Water:strength NA NA NA NA
## Superplasticizer49:Water:strength 5.478e-02 6.378e+01 0.001 0.999
## Superplasticizer50:Water:strength 4.003e-07 6.378e+01 0.000 1.000
## Superplasticizer51:Water:strength NA NA NA NA
## Superplasticizer52:Water:strength NA NA NA NA
## Superplasticizer53:Water:strength NA NA NA NA
## Superplasticizer54:Water:strength 2.006e-02 6.379e+01 0.000 1.000
## Superplasticizer55:Water:strength NA NA NA NA
## Superplasticizer56:Water:strength NA NA NA NA
## Superplasticizer57:Water:strength NA NA NA NA
## Superplasticizer58:Water:strength 4.896e-01 6.379e+01 0.008 0.994
## Superplasticizer59:Water:strength 4.003e-07 6.378e+01 0.000 1.000
## Superplasticizer60:Water:strength -6.098e+00 6.550e+01 -0.093 0.926
## Superplasticizer61:Water:strength NA NA NA NA
## Superplasticizer62:Water:strength -8.897e-01 6.383e+01 -0.014 0.989
## Superplasticizer63:Water:strength -1.574e-01 6.379e+01 -0.002 0.998
## Superplasticizer64:Water:strength NA NA NA NA
## Superplasticizer65:Water:strength NA NA NA NA
## Superplasticizer66:Water:strength NA NA NA NA
## Superplasticizer67:Water:strength NA NA NA NA
## Superplasticizer68:Water:strength NA NA NA NA
## Superplasticizer69:Water:strength 4.003e-07 6.378e+01 0.000 1.000
## Superplasticizer70:Water:strength 1.108e+00 6.380e+01 0.017 0.986
## Superplasticizer71:Water:strength 3.390e-02 6.378e+01 0.001 1.000
## Superplasticizer72:Water:strength -4.893e-01 6.379e+01 -0.008 0.994
## Superplasticizer73:Water:strength NA NA NA NA
## Superplasticizer74:Water:strength NA NA NA NA
## Superplasticizer75:Water:strength -1.565e-01 6.379e+01 -0.002 0.998
## Superplasticizer76:Water:strength -7.863e-03 6.379e+01 0.000 1.000
## Superplasticizer77:Water:strength NA NA NA NA
## Superplasticizer78:Water:strength NA NA NA NA
## Superplasticizer79:Water:strength -5.793e-02 6.378e+01 -0.001 0.999
## Superplasticizer80:Water:strength NA NA NA NA
## Superplasticizer81:Water:strength NA NA NA NA
## Superplasticizer82:Water:strength NA NA NA NA
## Superplasticizer83:Water:strength NA NA NA NA
## Superplasticizer84:Water:strength NA NA NA NA
## Superplasticizer85:Water:strength NA NA NA NA
## Superplasticizer86:Water:strength NA NA NA NA
## Superplasticizer87:Water:strength 4.003e-07 6.397e+01 0.000 1.000
## Superplasticizer88:Water:strength NA NA NA NA
## Superplasticizer89:Water:strength NA NA NA NA
## Superplasticizer90:Water:strength NA NA NA NA
## Superplasticizer91:Water:strength -1.856e-01 6.381e+01 -0.003 0.998
## Superplasticizer92:Water:strength NA NA NA NA
## Superplasticizer93:Water:strength NA NA NA NA
## Superplasticizer94:Water:strength NA NA NA NA
## Superplasticizer95:Water:strength NA NA NA NA
## Superplasticizer96:Water:strength NA NA NA NA
## Superplasticizer97:Water:strength NA NA NA NA
## Superplasticizer98:Water:strength NA NA NA NA
## Superplasticizer99:Water:strength NA NA NA NA
## Superplasticizer100:Water:strength NA NA NA NA
## Superplasticizer101:Water:strength NA NA NA NA
## Superplasticizer102:Water:strength NA NA NA NA
## Superplasticizer103:Water:strength NA NA NA NA
## Superplasticizer104:Water:strength NA NA NA NA
## Superplasticizer105:Water:strength NA NA NA NA
## Superplasticizer106:Water:strength NA NA NA NA
## Superplasticizer107:Water:strength NA NA NA NA
## Superplasticizer108:Water:strength NA NA NA NA
## Superplasticizer109:Water:strength NA NA NA NA
## Superplasticizer110:Water:strength NA NA NA NA
##
## Residual standard error: 52.56 on 753 degrees of freedom
## Multiple R-squared: 0.4934, Adjusted R-squared: 0.3077
## F-statistic: 2.657 on 276 and 753 DF, p-value: < 2.2e-16The Shapiro-Wilk normality test returns a p-value which is very small (less than 0.1) which shows that the data is normally distributed.The Normal Q-Q plot conforms normality for the residuals.
The scatter plot however indicates some trend i.e. there is a uniform decrease in the residual values over the entire dynamic range.This is opposite to our earlier results so the data might not be considered a very good fit. There are essentially 2 possibilities: 1.The slope of the regression line was computed incorrectly since the data shows trend. 2.The intercept of the regression line might or might not have been computed correctly.
The final analysis of fitting a model with different slopes for each factor level is again done to check the appropriateness of our initial analysis. We see from the results that none of the Superplasticizer:strength values give statistically significant results (all of them have p value greater than 0.05), which means our analysis was appropriate.Same is the case with Superplasticizer:Water probability values (not significant) indicating validity of our result.
References
Wikipedia
Data Source: https://archive.ics.uci.edu/ml/machine-learning-databases/concrete/compressive/