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MATH253 statistic
Creation date：2024-06-08 15:37:04
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Tag MATH253
statistic

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MATH253

1. Denote by µA, µB, µC , µD, µE the mean salt absorption for pasta types A,B,C,D,E respectively.
Then we test the null hypothesis H0 : µA = µB = µC = µD = µE against the alternative
H1 : µ values not all equal.
library(readxl)
salt <- read_excel("Tutorial8_salt.xlsx")
A <- salt$A
B <- salt$B
C <- salt$C
D <- salt$D
E <- salt$E
combined <- data.frame(cbind(A, B, C, D, E))
stacked <- stack(combined)
stacked
anovaresults <- aov(values ~ ind, data = stacked)
summary(anovaresults)
Note: When typing the symbol ∼ make sure that you type it using the appropriate key on your
keyboard. If you copy the symbol from the tutorial sheet and paste into R, R will not recognise
the symbol and the code won’t work.
R output:
Df Sum Sq Mean Sq F value Pr(>F)
ind 4 554.9 138.71 3.39 0.0211 *
Residuals 30 1227.4 40.91
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
The ANOVA table computes the test statistic F = 3.39. R computes the p-value as p = 0.0211.
Since 0.02 < p < 0.05, we can reject H0 at the 5% level but not at 2%. That is, we have evidence
that there are differences in mean salt absorption between the five types of pasta.
Error variance may be estimated as σ̂2 = MSE = 40.91, from the ANOVA table.
2. res <- residuals(anovaresults)
res
#histogram of residuals
hist(res, main = paste("Histogram of residuals"), breaks = 10, xlab = "Residuals",
xlim = range(-15,20), col="purple")
#normality test - Anderson-Darling test
library(nortest)
ad.test(res)
#normal probability plot
qqnorm(res, col="blue")
qqline(res, col="red")
R output:
1
Anderson-Darling normality test
data: res
A = 0.096011, p-value = 0.9965
The histogram of residuals looks reasonably like the symmetrical bell-shaped curve of a normal
distribution (although there is a slight skew). The points on the normal probability plot look
reasonably close to a straight line. We perform the Anderson-Darling normality test. We test
H0 : errors follow a normal distribution vs H1 : errors do not follow a normal distribution. The
p-value is p = 0.9965 suggesting that we cannot reject H0 even at the 99% level and so there is
no evidence against normality. Hence it does seem reasonable to assume normality of the errors.
3. #tests for equal variances
bartlett.test(values ~ ind, data = stacked)
library(car)
leveneTest(values ~ ind, data = stacked)
#boxplot
means <- c(mean(A), mean(B), mean(C), mean(D), mean(E))
means
boxplot(A, B, C, D, E, col = "lightblue", names=c("A", "B", "C", "D", "E"))
points(means)
R output:
Bartlett test of homogeneity of variances
data: values by ind
Bartlett’s K-squared = 6.0273, df = 4, p-value = 0.1971
Levene’s Test for Homogeneity of Variance (center = median)
Df F value Pr(>F)
group 4 1.0505 0.3979
30
2
Denote by σA, σB, σC , σD, σE the standard deviations of salt absorption for pasta typeA,B,C,D,E
respectively. Both Bartlett’s and Levene’s tests are testing the null hypothesis H0 : σA = σB =
σC = σD = σE against the alternative H1 : σ values not all equal. R computes p = 0.1971 for
Bartlett’s test and p = 0.3979 for Levene’s test. In both cases, we cannot reject H0 even at the
10% level. That is, there is no evidence of differences in standard deviations between the five
types of pasta. The assumption of equal variances appears justified for these data.
Note that it would be appropriate to perform only Bartlett’s test here because in Part 2 we have
concluded that it seems that the normality assumption is satisfied.
The boxplots do not show variance (or standard deviation) explicitly, but some measure of spread
is given by the inter-quartile range, shown by the length of the box in each plot. All five boxes
are about the same length as each other (the boxes for Pasta A and C are noticeably longer
than the others, but it’s not a very big difference). This fits in with our conclusion that all five
distributions have about the same variance.
A rough rule is that the assumption of equal standard deviations is OK provided the ratio
between largest and smallest standard deviation is smaller than 2. For our data, the largest SD
is σA = 9.335034 and the smallest SD is σB = 3.450328. (You can use the command sd to find
the standard deviation for each group.) The ratio is σA/σB = 9.335034/3.450328 = 2.706 > 2.
Hence the rough rule suggests that the variances are not equal. This contradicts the conclusion
from the formal tests.
4. #post-hoc tests
TukeyHSD(anovaresults)
library(PMCMRplus)
summary(lsdTest(anovaresults))
R output:
Tukey multiple comparisons of means
95% family-wise confidence level
Fit: aov(formula = values ~ ind, data = stacked)
$ind
diff lwr upr p adj
B-A 7.428571 -2.488707 17.345849 0.2174730
C-A 3.142857 -6.774421 13.060135 0.8872327
D-A -4.142857 -14.060135 5.774421 0.7447137
E-A -1.428571 -11.345849 8.488707 0.9932777
3
C-B -4.285714 -14.202992 5.631564 0.7206802
D-B -11.571429 -21.488707 -1.654151 0.0158914
E-B -8.857143 -18.774421 1.060135 0.0977396
D-C -7.285714 -17.202992 2.631564 0.2337999
E-C -4.571429 -14.488707 5.345849 0.6709102
E-D 2.714286 -7.202992 12.631564 0.9303811
Pairwise comparisons using Least Significant Difference Test
data: values by ind
alternative hypothesis: two.sided
P value adjustment method: none
H0
t value Pr(>|t|)
B - A == 0 2.173 0.0378317 *
C - A == 0 0.919 0.3653109
D - A == 0 -1.212 0.2350836
E - A == 0 -0.418 0.6790478
C - B == 0 -1.253 0.2197098
D - B == 0 -3.384 0.0020041 **
E - B == 0 -2.591 0.0146515 *
D - C == 0 -2.131 0.0414062 *
E - C == 0 -1.337 0.1912563
E - D == 0 0.794 0.4335032
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
Fisher method suggests that the mean salt absorption of pasta B is significantly different than
that of A (at 5%), E (at 5%) and D (at 1%); also, the mean salt absorption of pasta D is
significantly different at 5% than that of C. So, there is evidence of differences in the mean salt
absorption of pasta between groups B and A, B and E, C and D. There is strong evidence of
differences in the mean salt absorption of pasta between groups B and D. There is no evidence
of differences between other pairs of groups.
Tukey method suggests that only groups B and D differ significantly at 5% and groups B and
E at 10%. There is evidence of differences in the mean salt absorption of pasta between groups
B and D. There is weak evidence of differences in the mean salt absorption of pasta between
groups B and E. There is no evidence of differences between other pairs of groups.
The difference in results arises because Tukey’s method uses an adjustment for multiple com-
parisons and Fisher method does not. That is why it is recommended to use Tukey’s HSD test
when using R.
PART B
1. library(readxl)
vision <- read_excel("Tutorial8_vision.xlsx")
G1 <- vision$Group1
G2 <- vision$Group2
G3 <- vision$Group3
combinedB <- data.frame(cbind(G1, G2, G3))
stackedB <- stack(combinedB)
4
stackedB
anovaB <- aov(values ~ ind, data = stackedB)
summary(anovaB)
R output:
Df Sum Sq Mean Sq F value Pr(>F)
ind 2 227.0 113.52 6.817 0.00222 **
Residuals 57 949.2 16.65
---
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
Denote by µ1, µ2, µ3 the mean VA for three groups, no diabetic retinopathy, early diabetic
retinopathy and late diabetic retinopathy, respectively. Then we test the null hypothesis H0 :
µ1 = µ2 = µ3 against the alternative H1 : µ values not all equal.
R computes the p-value as p = 0.00222. Since p < 0.01, we can reject H0 at the 1% level but
not at 0.1%. That is, we have strong evidence that there are differences in mean visual acuity
between the three stages of diabetic retinopathy.
2. resB <- residuals(anovaB)
resB
hist(resB, main = paste("Histogram of residuals"), breaks = 10, xlab = "Residuals",
xlim = range(-11,11), col="purple")
#normality test - Anderson-Darling test
library(nortest)
ad.test(resB)
#normal probability plot
qqnorm(resB, col="blue")
qqline(resB, col="red")
R output:
Anderson-Darling normality test
data: resB
A = 0.27225, p-value = 0.6583
5
The points on the normal probability plot look reasonably close to a straight line.
We test H0 : errors follow a normal distribution vs H1 : errors do not follow a normal distribution.
The p-value is p = 0.6583 suggesting that we cannot reject H0 even at the 65% level and so there
is no evidence against normality. Hence it seems that the normality assumption is justified.
#equal variances test
bartlett.test(values ~ ind, data = stackedB)
R output:
Bartlett test of homogeneity of variances
data: values by ind
Bartlett’s K-squared = 0.066925, df = 2, p-value = 0.9671
Since it seems that the normality assumption of ANOVA is satisfied, we use Bartlett’s test to
check the assumption of equal variances. Denote by σ21, σ
2
2, σ
2
3 the variances of VA for diabetic
retinopathy groups 1, 2, 3 respectively. We are testing the null hypothesis H0 : σ
2
1 = σ
2
2 = σ
2
3
against the alternative H1 : σ
2 values not all equal. R computes p = 0.9671 for Bartlett’s test.
Hence, we cannot reject H0 even at the 96% level. That is, there is no evidence of differences
in variances between the three groups of patients. The assumption of equal variances appears
justified for these data.
3. TukeyHSD(anovaB)
library(PMCMRplus)
summary(lsdTest(values ~ ind, data = stackedB))
R output:
Tukey multiple comparisons of means
95% family-wise confidence level
Fit: aov(formula = values ~ ind, data = stackedB)
$ind
diff lwr upr p adj
G2-G1 -2.70 -5.805282 0.4052823 0.1004289
G3-G1 -4.75 -7.855282 -1.6447177 0.0014807
G3-G2 -2.05 -5.155282 1.0552823 0.2587838
Pairwise comparisons using Least Significant Difference Test
data: values by ind
alternative hypothesis: two.sided
P value adjustment method: none
H0
t value Pr(>|t|)
G2 - G1 == 0 -2.092 0.04087170 *
G3 - G1 == 0 -3.681 0.00051741 ***
G3 - G2 == 0 -1.589 0.11767511
---
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
6
Fisher’s LSD method suggests that the mean visual acuity of Group 1 is significantly different
from that of 2 (at 5%) and 3 (at 0.1%); but the mean visual acuity between groups 2 and 3 is not
significantly different at the 11% level. There is evidence that the mean visual acuity between
groups 1 and 2 is different. There is very strong evidence that the mean visual acuity between
groups 1 and 3 is different. There is no evidence that the mean visual acuity between groups 2
and 3 is different.
Tukey’s HSD method suggests that the mean visual acuity of groups 1 and 3 differ significantly
(at 0.2%) and the mean visual acuity of groups 1 and 2 is not significantly different at the 10%
level. The mean visual acuity between groups 2 and 3 is not significantly different even at the
25% level. There is no evidence that the mean visual acuity between groups 1 and 2 is different.
There is strong evidence that the mean visual acuity between groups 1 and 3 is different. There
is no evidence that the mean visual acuity between groups 2 and 3 is different.
4. meansB <- c(mean(G1), mean(G2), mean(G3))
meansB
boxplot(G1, G2, G3, col = "purple", main = "Boxplots",
names=c("Group 1", "Group 2", "Group 3"), boxwex=0.4)
points(meansB)
R output:
There is not a significant difference in the mean VA between groups 2 and 3 (i.e. early and late
stage of diabetic retinopathy) according to Tukey’s or Fisher’s tests.
Also there is a lot of overlap of the VA data between groups 2 and 3 in the box plot.
This means that we cannot use VA to differentiate between the early stage and late stage of
diabetic retinopathy.
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270M CIS 3207 Programs CSCI 2132 CS270 CSE-381 COMP3230A CIS 212 CSI 333 FINM 326 FINM-36702 Introduction to Software Analysis EEE102 GNG1106 ECS 60 CS 161 CSCI851 CSE1222 EDPS0249 TCHR5003 ANTA02 BBUS1SBY SOSC 1140 MDIA2091 FINS3641 INSY 661 ITEC3040 CPSC 481 computer DEE3519 COMP304 CSE536 Microtheory OFFICE 280 CIVE50003 COSC2673 ECS784U CMPSC/MATH 455 AMA 505 CSE 158 BUS1040 FIT3179 FIT5202 Math 104A Math 111A MBUS107 COMPSCI 761 COMP3710 COMPSCI 316 COMP559 COMP557 CP1 1902 COMP 310 ECED 3403 COMPX523 AGCM 3103 EMET 7001 ENGR3821 CISC 124 IPC comp20005 CS 323 CS105 STAT 326 MATH4007 CS4551 COMP1000 ACIT 4640 ENVX3002 CS2034 FC712 algorithms CSCI 4155 CSI4142 POL 850 MATH20811 2B03 CSCI803 Economies COMP 2016 Architect TRADE INFO1110 FIT1043 networks Robotics ICS 31 ECEC-301 FIT2099 CS 3640 CSE1010 Computational COMP1001 CSE 231 CS10 FIT5166 CSE 400 CSCI561 Math 350 CO-496 ENGG6400 CSCI 1133 COMP3055 MATH 819 CSE 6363 CIS 555 CS4420 COMP0037 ECS 140A Spanning Tree Protocol network 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EBU5376 STAT6118 ECON 20110 COMU7000 average acceleration BUSI4412 BFF3121 BIT233 ECMM462 ETC3430 DESN2348 INTE2584 ENME502 MA3XJ INTE2627 MA2815 MA3AM L0101 Math 3MB3 CPCCBC4004 EEEM061 0502E220 APPLIED 6SSMN310 ENV 3310 PM2PCOL3 CONS6201 MATH502 FIT5003 EEEN313 MATH3001 MECH3460 ECON30009 COSC1252 Assembler COMP 3023 ENGG7601 COMP20005 COMP SCI 1103 ICSI 402 INF2C CS320 COMPSCI7064 ECE 3220 PROGRAMMAMING MIPS Computer Science 2413 AM3611 MSOA TE2101 FFT CMPSC-121 CSE 109 COMP SCI 7064 HCI CMP-5013A CCN2042 COMP3400 EEEN30052 CO4215 EE101 KIT107 CSE 421 CSE521 MCD4720 EECS 280 MIT 403 GRPC VG101 APiE EECE 1080C FIT3143 CSE 325 ISOM 3029 CS3307 State SIT255 Engineering MQF Modelling ACX3150 SIA322 ACCT90004 COMU7201 MGMT90018 WRIT1000 COMP3702 BISM1201 COMP7505 IDEA9106 CSE3SMT BFC5926 BFC5280 BFC5935 QBUS3330 ECON3700 IFN521 ECON2121 COSC2626 MA413 DDES9903 ENGG1300 COMP9337 CO2201 ACCT2002 ICM289 MMM054 MSBA-204 DIGM707 CSC2250 Econ312 EVAT python CAN404 LUBS2840 CSCI-GA.2250 COMSM0093 COMP281 AFM 333 IBUS1001 CP2501 Math 105A VE320 TEC101 SOCY2166 COM240 EFIM20033 ENGI4467 EFIM20034 EC 979 BE432 7EJ502 CSSE6400 COMP90025 Econ 312 ENG2031 ECO0006 EXERSCI 207 CAN209 ECO 137W FT312 ECO 2199 MATH32052 BDW4213 APS 360 SAS 6 CHME0017 PO92Q Economics 01AV MATH39512 MATH 5486 COMP8620 ISE102 ECON235 ECON 206 FI 345 PHAY0020 MATH0099 MATH0085 EFB106 PHY 134 MAE115 circuits Chemistry CSE12 MATH-UA 233 PHAS0038 COMP2140 CONMGNT 7049 COMP151101 ECON 485 ACX5903 ENGN6536 ARCH1162 Law PHAS0042 CCF1C03 MATH 351 Phys 139 AG942 MGT001371 APH106 ECON1051 MA4AM MA3Z7 NUMBER THEORY REST0006 SESS0026 STAT0023 TABL 2741 CMP4008Y EFIM10015 Acct 560 PSYC735 GEOG 5 ACCT3104 CSEC5616 PHIL1012 ITLS6111 ECON8022 5CCM226A compX123 MA4XJ R001 EWB LLP120 COMP643 ENG 103 ECON 23950 ACCT 207 FINC 616 Bio99 Math 137A Psychology BCPM0054 MATH20212 EC3450 CENV6141 PSY 205 CE335 CS365 Calculus ACSD INFT6800 MCIT INFO6030 PHYS 111LV H63EEJ ECON2151 ENGI 2181 MTRX1701 PLIT08009 CS240 Math 270 CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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