Zkouška z MAS02, 7. 9. 2021
Popis situace: U 23 českých profesionálních basketbalistů byly zjišťovány hodnoty těchto proměnných:
vyska ... tělesná výška v cm hmotnost ... tělesná hmotnost v kg
tuk ... procento tělesného tuku (u basketbalistů by se tyto hodnoty měly pohybovat od 7 % do 13 %)
V02_max ... aerobní kapacita (maximální množství kyslíku za minutu, které může organismus využít při intenzívním fyzickém zatížení; jedná se o ukazatel tělesné zdatnosti; ideální hodnota by se měla pohybovat kolem 60 ml/kg)
Údaje jsou uloženy v souboru basketbal.csv.
Cíle výzkumu:
1. Pomocí metod korelační analýzy prozkoumat závislosti mezi proměnnými.
2. Sestavit model vícenásobné lineární regrese, který umožní popsat závislost aerobní kapacity na výšce, hmotnosti a procentu tělesného tuku.
Upozornění: Pro ověření vícerozměrné normality dat je použita funkce CM.test z knihovny mvtnorm.
Výstupy ze systému R
> 1 ibrary(mvtnorm)
> CM.test(basketbal,qqplot=T)
Cramer-von Mises test for Multivariate Normality
data : basketbal
CM : 0.07362245
p-value : 0.5191481
Result    : Data are multivariate normál  (sig.level = 0.05)
Chi-Square Q-Q Plot
n i i i r
2 4 6 S 10
Squared Mahalanobis Distance
> cor.test(basketbal$V02_max,basketbal$vyska)
Pearson's product-moment correlation
data:    basketbal$V02_max and basketbal$vyska t = -2.7973, df = 21, p-value = 0.01079
alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval:
-0.7682332 -0.1385822 sample estimates: cor
-0.5210216
> cor.test(basketbal$V02_max,basketbal$hmotnost)
Pearson's product-moment correlation
data:    basketbal$V02_max and basketbal$hmotnost
t = -10.194, df = 21, p-value = 1.381e-09
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.9624444 -0.8010513 sample estimates: cor
-0.9120869
> cor.test(basketbal$V02_max,basketbal$tuk)
Pearson's product-moment correlation
data:    basketbal$V02_max and basketbal$tuk t = -5.108, df = 21, p-value = 4.646e-05
alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval:
-0.8850136 -0.4791772 sample estimates: cor
-0.7443558
> cor.test(basketbal$vyska,basketbal$hmotnost)
Pearson's product-moment correlation
data:    basketbal$vyska and basketbal$hmotnost t = 4.0081, df = 21, p-value = 0.0006373
alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval:
0.3378260 0.8420445 sample estimates: cor
0.6583511
> cor.test(basketbal$vyska,basketbal$tuk)
Pearson's product-moment correlation
data:    basketbal$vyska and basketbal$tuk t = 1.7241, df = 21, p-value = 0.09939
alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval:
-0.07027838 0.66744879 sample estimates: cor
0.3521245
> cor.test(basketbal$hmotnost,basketbal$tuk)
Pearson's product-moment correlation
data:    basketbal$hmotnost and basketbal$tuk t = 4.0019, df = 21, p-value = 0.0006467
alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval:
0.3369272 0.8417491 sample estimates:
cor 0.657776
pcor(basketbal [-1])
$estimate
hmotnost tuk
V02_max
hmotnost 1.00000000 -0.07721672
tuk
-0.07721672 1.00000000
-0.83992326 -0.46761575
V02_max -0.8399233 -0.4676157 1.0000000
$p.value
hmotnost tuk V02_max
hmotnost 0.000000e+00 0.73269113 1.011868e-06 tuk 7.326911e-01 0.00000000 2.820084e-02
V02_max   1.011868e-06 0.02820084 0.000000e+00
$statistic
hmotnost tuk V02_max
hmotnost 0.0000000 -0.3463578 -6.921347 tuk -0.3463578   0.0000000 -2.365840
V02_max   -6.9213471 -2.3658399 0.000000
$n
[1] 23
$gp [1] 1
$method
[1] "pearson"
> pcor (basketbal [-; $estimate
vyska
vyska 1.0000000 tuk -0.0626353 V02_max -0.4142560
])
tuk V02_max -0.0626353 -0.4142560 1.0000000 -0.7020999 -0.7020999 1.0000000
$p.value
vyska tuk vo2_max
vyska 0.00000000 0.7818504001 0.0552699949 tuk 0.78185040 0.0000000000 0.0002701834 V02_max 0.05526999 0.0002701834 0.0000000000
$statistic
vyska tuk vo2_max
vyska 0.0000000 -0.2806647 -2.035475 tuk -0.2806647 0.0000000 -4.409467 V02_max -2.0354755 -4.4094672 0.000000
$n
[1] 23
$gp [1] 1
$method
[1] "pearson"
> pcor(basketbal [-3]) $estimate
vyska
vyska 1.0000000 hmotnost 0.5233145 V02_max 0.2574508
hmotnost 0.5233145
V02_max 0.2574508
1.0000000 -0.8857539 -0.8857539 1.0000000
$p.value
vyska 0.00000000 1.244393e-02 2 hmotnost 0.01244393 0.000000e+00 4 V02_max   0.24739789 4.233245e-08 0.000000e+00
vyska hmotnost 0.00000000 1.244393e-02
V02_max ,473979e-01 ,233245e-08
$statistic
vyska
vyska 0.000000 hmotnost 2.746421 V02_max 1.191520
hmotnost 2.746421
V02_max 1.191520
0.000000 -8.534241 -8.534241 0.000000
$n
[1] 23
$gp [1] 1
$method
[1] "pearson"
pcor(basketbal)
$estimate
vyska
hmotnost
tuk
V02_max
vyska 1.00000000 0.52105379
hmotnost               tuk V02_max
0.52105379 -0.02616148 0.2179797
1.00000000 -0.05225222 -0.8132347
-0.02616148 -0.05225222    1.00000000 -0.4505123
0.21797971 -0.81323473 -0.45051226 1.0000000
$p.value
vyska        hmotnost tuk V02_max
vyska 0.0000000 1.543230e-02 0.91037573 3.425065e-01 hmotnost 0.0154323 0.000000e+00 0.82202548 7.399455e-06 tuk 0.9103757 8.220255e-01 0.00000000 4.041368e-02
V02_max   0.3425065 7.399455e-06 0.04041368 0.000000e+00
$statistic
vyska hmotnost tuk
V02_max
vyska 0.0000000 2.6609928 -0.1140743 0.9735625
hmotnost 2.6609928 0.0000000 -0.2280737 -6.0914074
tuk
-0.1140743 -0.2280737 0.0000000 -2.1996000
V02_max 0.9735625 -6.0914074 -2.1996000 0.0000000
$n
[1] 23
$gp [1] 2
$method
[1] "pearson"
> model<-
1 m(basketbal$V02_max~basketbal$vyska+basketbal$hmotnost+basketbal $tuk)
> summary(model)
Call :
lm(formula = basketbal$V02_max ~ basketbal$vyska + basketbal$hmotnost + basketbal$tuk)
Residuals:
Min 1Q   Median 3Q Max
-2.6811 -1.0552 -0.1376   0.6958 4.0287
Coefficients:
Estimate Std. Error t value Pr(>|t|) (intercept) 79.60140       7.28196   10.931 1.23e-09 ***
basketbal$vyska        0.04633      0.04759     0.974 0.3425 basketbal$hmotnost -0.31683      0.05201   -6.091 7.40e-06 *** basketbal$tuk -0.33671      0.15308   -2.200     0.0404 *
Signif. codes:   0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 '  ' 1
Residual standard error: 1.943 on 19 degrees of freedom Multiple R-squared:   0.8749,    Adjusted R-squared: 0.8551 F-statistic: 44.29 on 3 and 19 DF,    p-value: 9.031e-09
> model1<-1m(basketbal$V02_max~basketbal$hmotnost+basketbal$tuk)
> summary(model 1)
Call :
lm(formula = basketbal$V02_max ~ basketbal$hmotnost + basketbal$tuk)
Residuals:
Min 1Q   Median 3Q Max
-3.0164 -1.1064 -0.3972    1.1637 4.2497
Coefficients:
Estimate Std. Error t value Pr(>|t|) (intercept) 85.96030       3.21533   26.735   < 2e-16 ***
basketbal$hmotnost -0.28617      0.04135   -6.921 1.01e-06 *** basketbal$tuk -0.35798      0.15131   -2.366     0.0282 *
Signif. codes:   0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 '  ' 1
Residual standard error: 1.94 on 20 degrees of freedom Multiple R-squared:   0.8687,    Adjusted R-squared: 0.8555 F-statistic: 66.14 on 2 and 20 DF,    p-value: 1.528e-09
> anova(model , modeli) Analysis of Variance Table
Model 1: basketbal$V02_max ~ basketbal$vyska + basketbal$hmotnost + basketbal$tuk
Model 2: basketbal$V02_max ~ basketbal$hmotnost + basketbal$tuk
Res.Df      RSS Df Sum of Sq F Pr(>F)
1 19 71.694
2 20 75.271 -1     -3.5765 0.9478 0.3425
> par(mfrow=c(2,2))
> piot(model 1)
Residuals vs Fitted
Normal Q-Q
Fitted values
Theoretical Quantiles
Scale-Location
55 ° S- o
	2t>	150
	O 0 o	0
í^"~~~~~"~"——__	0	
		O 0 0 o
I 45	I I 60 55 Fitted values	1 60
> shapi ro.test(model 1$resi dual s)
Residuals vs Leverage
—   Cook's distance
0.00  0.05  0.10  0.15  0.20  0.25  0.30 0.35 Leverage
Shapiro-Wilk normality test
data: modell$residuals
W = 0.96952, p-value = 0.6774
> library(car)
> durbinWatsonTest(model 1)
lag Autocorrelation D-W Statistic p-value 1 0.02514025 1.927389 0.73
Alternative hypothesis: rho != 0
> confint(model 1)
2.5 % 97.5 %
(intercept) 79.2532493 92.66735262
basketbal$hmotnost -0.3724219 -0.19992679 basketbal$tuk -0.6736194 -0.04234917
> vi f(modeli)
basketbal$hmotnost basketbal$tuk
1.76264 1.76264
> (MAPE<-100 * mean(abs(modell$residuals/basketbal$V02_max))) [1] 2.753056