Exercises from Units 1-8

Lesson 1: Measures of Central Tendency, Dispersion and Association

• Example 1-5: Women’s Health Survey (Descriptive Statistics)

  Variable	n	Mean	StdDev	Minimum	Maximum
  calcium	737	624.04925	397.277540	7.44	2866.440
  iron	737	11.12990	5.984191	0.00	58.668
  protein	737	65.80344	30.575756	0.00	251.012
  a	737	839.63535	1633.539828	0.00	34434.270
  c	737	78.92845	73.595272	0.00	433.339

• The variance-covariance matrix :

  	calcium	iron	protein	vitamin A	vitamin C
  calcium	157829.4439	940.08944	6075.8163	102411.127	6701.6160
  iron	940.0894	35.81054	114.0580	2383.153	137.6720
  protein	6075.8163	114.05803	934.8769	7330.052	477.1998
  vitamin A	102411.1266	2383.15341	7330.0515	2668452.371	22063.2486
  vitamin C	6701.6160	137.67199	477.1998	22063.249	5416.2641

• Sample correlations

  	calcium	iron	protein	vitamin A	vitamin C
  calcium	1.0000000	0.3954301	0.5001882	0.1578060	0.2292111
  iron	0.3954301	1.0000000	0.6233662	0.2437905	0.3126009
  protein	0.5001882	0.6233662	1.0000000	0.1467574	0.2120670
  vitamin A	0.1578060	0.2437905	0.1467574	1.0000000	0.1835227
  vitamin C	0.2292111	0.3126009	0.2120670	0.1835227	1.0000000

• Example 1-6: Woman’s Health Survey (Variance)

Variable	Variance
Calcium	157829.443870502
Iron	35.810535582296
Protein	934.87688135283
Vitamin A	2668452.37064258
Vitamin C	5416.26407815779
Total	2832668.76600817

• Example 1-7: Woman’s Health Survey (Generalized Variance)

Example: Nutrient Intake Data - Generalized variance

	calcium	iron	protein	vitamin A	vitamin C
calcium	157829.4439	940.08944	6075.8163	102411.127	6701.6160	2.831042e+19
iron	940.0894	35.81054	114.0580	2383.153	137.6720	2.831042e+19
protein	6075.8163	114.05803	934.8769	7330.052	477.1998	2.831042e+19
vitamin A	102411.1266	2383.15341	7330.0515	2668452.371	22063.2486	2.831042e+19
vitamin C	6701.6160	137.67199	477.1998	22063.249	5416.2641	2.831042e+19

Lesson 2: Linear Combinations of Random Variables

• 2-3: Women’s Health Survey (Population Mean)

Variable	Mean
Calcium	624.04925
Iron	11.12990
Protein	65.80344
Vitamin A	839.63535
Vitamin C	78.92845

[1] "Estimated mean intake of the two vitamins:"

[1] 79.76808

• 2-4: Women’s Health Survey (Population Variance)

	calcium	iron	protein	vitamin A	vitamin C
calcium	157829.4439	940.08944	6075.8163	102411.127	6701.6160
iron	940.0894	35.81054	114.0580	2383.153	137.6720
protein	6075.8163	114.05803	934.8769	7330.052	477.1998
vitamin A	102411.1266	2383.15341	7330.0515	2668452.371	22063.2486
vitamin C	6701.6160	137.67199	477.1998	22063.249	5416.2641

[1] "The sample variance of Y is:"

[1] 5463.059

• 2-5: Women’s Health Survey (Pop. Covariance and Correlation)

	calcium	iron	protein	vitamin A	vitamin C
calcium	157829.4439	940.08944	6075.8163	102411.127	6701.6160
iron	940.0894	35.81054	114.0580	2383.153	137.6720
protein	6075.8163	114.05803	934.8769	7330.052	477.1998
vitamin A	102411.1266	2383.15341	7330.0515	2668452.371	22063.2486
vitamin C	6701.6160	137.67199	477.1998	22063.249	5416.2641

Note, \(Y_1\) is the total intake of vitamins A and C and \(Y_2\) is the total intake of calcium and iron.

[1] "The sample covariance between Y_1 and Y_2:"

[1] 6944.082

[1] "The sample variance of Y_2 is:"

[1] 159745.4

[1] "The sample correlation between Y_1 and Y_2 is:"

[1] 0.2350621

Lesson 3: Graphical Display of Multivariate Data

Graphical Method

• Univariate Cases (Histogram)

• Bivariate Cases (Scatter Plots)

Correlation Matrix (Quarter Root Transformations)

	S_S_calc	S_S_iron	S_S_prot	S_S_a	S_S_c
S_S_calc	1.0000000	0.5035233	0.5719948	0.5159994	0.3543776
S_S_iron	0.5035233	1.0000000	0.7355742	0.5028902	0.4214321
S_S_prot	0.5719948	0.7355742	1.0000000	0.4271488	0.3334733
S_S_a	0.5159994	0.5028902	0.4271488	1.0000000	0.4313927
S_S_c	0.3543776	0.4214321	0.3334733	0.4313927	1.0000000

• Trivariate Cases (Rotating Scatter Plots )

• Multivariate Cases (Matrix of Scatter Plots)

Lesson 4: Multivariate Normal Distribution

• Bivariate Normal Distribution

• 4.3 Calculating Mahalanobis Distance

  dist2
  0.1546640
  8.9860868
  3.8413545
  2.8801184
  0.6785061
  0.7314023
  0.6988023
  1.1029161
  37.6264510
  1.0880096
  0.6447859
  0.8679835
  0.2064665
  0.0794967
  0.2787355
  4.6633784
  12.1952311
  6.0454561
  0.2917265
  1.0271777
  4.6434462
  0.8539685
  2.1987106
  1.5010624
  1.2807514
  2.8544397
  1.3753355
  0.9072836
  10.5129471
  5.7833057

• 4.7 - Example: Wechsler Adult Intelligence Scale

Variable	Mean
Information	12.567568
Similarities	9.567568
Arithmetic	11.486486
Picture Completion	7.972973

	info	sim	arith	pict
info	11.474474	9.0855856	6.382883	2.0713213
sim	9.085586	12.0855856	5.938438	0.5435435
arith	6.382883	5.9384384	11.090090	1.7912913
pict	2.071321	0.5435435	1.791291	3.6936937

The eigenvalues and corresponding eigenvectors are given below:

Eigenvalue
26.245278
6.255366
3.931553
1.911647

	X1	X2	X3	X4
info	-0.6057467	-0.6047618	-0.5051337	-0.1103360
sim	0.2176473	0.4958117	-0.7946452	-0.2744802
arith	0.4605028	-0.3196759	-0.3349263	0.7573433
pict	0.6112591	-0.5350152	0.0346888	-0.5821664

Lesson 5: Sample Mean Vector and Sample Correlation and Related Inference Problems

• Example 5-4

  Variable	n	Mean	StdDev
  domradius	37	0.84380	0.1140245
  domhumerus	37	1.79268	0.2834735
  domulna	37	0.70440	0.1075566

• Confidence Interval

One at a time intervals:

Confidence Intervals

Variable	Lower	Upper
domradius	0.797	0.891
domhumerus	1.676	1.910
domulna	0.660	0.749

Bonferroni Method:

Simultaneous Confidence Intervals (Bonferroni Correction)

Variable	Lower	Upper
domradius	0.785	0.902
domhumerus	1.647	1.939
domulna	0.649	0.760

Simultaneous Confidence Region Method:

Confidence Intervals

Variable	Lower	Upper
domradius	0.772	0.916
domhumerus	1.614	1.972
domulna	0.636	0.772

Data Frame for Confidence Interval:

Obs	variable	TYPE	FREQ	n	xbar	s2	t1	tb	f	loone	upone	lobon	upbon	losim	upsim
1	domhumerus	0	25	25	1.79268	0.0803572	2.063899	2.573641	3.049125	1.6756679	1.9096921	1.6467682	1.9385918	1.6135844	1.9717756
1	domradius	0	25	25	0.84380	0.0130016	2.063899	2.573641	3.049125	0.7967330	0.8908670	0.7851084	0.9024916	0.7717605	0.9158395
1	domulna	0	25	25	0.70440	0.0115684	2.063899	2.573641	3.049125	0.6600028	0.7487972	0.6490376	0.7597624	0.6364469	0.7723531

• Example 5-5: Wechsler Adult Intelligence Scale

Correlations among variables in the table below:

	info	sim	arith	pict
info	1.0000000	0.7715297	0.5658260	0.3181637
sim	0.7715297	1.0000000	0.5129451	0.0813523
arith	0.5658260	0.5129451	1.0000000	0.2798777
pict	0.3181637	0.0813523	0.2798777	1.0000000

The correlation between Information and Similarities is 0.7715297.

T-test Statistic	7.174605

• Example 5-6: Fisher Transform and its 95% confidence interval

Metric	Value
Fisher Transform (z)	1.02409623292489
95% CI for z	( 0.688 , 1.3602 )
95% CI for r	( 0.5967 , 0.8764 )

This yields 95% confident that the interval (0.5967, 0.8764) contains the correlation between Information and Similarities scores.

Lesson 6: Multivariate Conditional Distribution and Partial Correlation

• 6-2: Wechsler Adult Intelligence Scale

Example 6-2: Wechsler Adult Intelligence Data

	info	sim	arith	pict
info	1.0000000	0.7118787	0.2267109	0.3463270
sim	0.7118787	1.0000000	0.1890167	-0.2962762
arith	0.2267109	0.1890167	1.0000000	0.1758075
pict	0.3463270	-0.2962762	0.1758075	1.0000000

The partial correlation between Information and Similarities is r=0.7118787.

	info	sim	arith	pict
info	Inf	1.0240963	0.6413618	0.3296027
sim	1.0240963	Inf	0.5667183	0.0815325
arith	0.6413618	0.5667183	Inf	0.2875493
pict	0.3296027	0.0815325	0.2875493	Inf

• Example 6-3: Wechsler Adult Intelligence Data

  T-test Statistic	5.822893

t

Fisher Transform (z)	0.890982564683951
95% CI for z	( 0.5445 , 1.2375 )
95% CI for r	( 0.4964 , 0.8447 )

Based on the result, we can conclude that we are 95% confident that the interval (0.4964, 0.8447) contains the partial correlation between Information and Similarities scores given scores on Arithmetic and Picture Completion.

Lesson 7: Inferences Regarding Multivariate Population Mean

• 7-1: Woman’s Survey Data (Hotelling’s \(T^2\) Test)

The sample variance-covariance matrix:

	calcium	iron	protein	vitamin A	vitamin C
calcium	157829.4439	940.08944	6075.8163	102411.127	6701.6160
iron	940.0894	35.81054	114.0580	2383.153	137.6720
protein	6075.8163	114.05803	934.8769	7330.052	477.1998
vitamin A	102411.1266	2383.15341	7330.0515	2668452.371	22063.2486
vitamin C	6701.6160	137.67199	477.1998	22063.249	5416.2641

Statistic	Value
T^2	1758.5413
F	349.7968
df1	5.0000
df2	732.0000
p	0.0000

• 7-2 to 7-4: Confidence Intervals

The one-at-a-time confidence intervals are given by “loone” and “upone”. On the other hand, simultaneous confidence intervals are given by the columns for “losim” and “upsim” while for Bonferroni intervals are given by “lobon” and “upbon”.

Variable	n	means	var	t1	tb	f	loone	upone	lobon	upbon	losim	upsim
calcium	737	624.04925	1.578294e+05	1.963192	2.582526	2.22634	595.32008	652.77843	586.25681	661.84169	575.09118	673.00733
iron	737	11.12990	3.581054e+01	1.963192	2.582526	2.22634	10.69715	11.56265	10.56063	11.69917	10.39244	11.86735
protein	737	65.80344	9.348769e+02	1.963192	2.582526	2.22634	63.59235	68.01453	62.89481	68.71207	62.03547	69.57141
vitamin A	737	839.63535	2.668452e+06	1.963192	2.582526	2.22634	721.50572	957.76498	684.23906	995.03163	638.32780	1040.94289
vitamin C	737	78.92845	5.416264e+03	1.963192	2.582526	2.22634	73.60640	84.25050	71.92743	85.92946	69.85901	87.99788

• Example 7-5: Women’s Health Survey (Profile Plots)

• 7-9: Spouse Data (Paired Hotelling’s)

            d1          d2         d3          d4
d1  0.82298851  0.07816092 -0.0137931 -0.05977011
d2  0.07816092  0.80919540 -0.2137931 -0.15632184
d3 -0.01379310 -0.21379310  0.5620690  0.51034483
d4 -0.05977011 -0.15632184  0.5103448  0.60229885

Test Statistics

Test	Value
T^2	13.1278403
F	2.9424470
df1	4.0000000
df2	26.0000000
p	0.0393691

• Example 7-10: Spouse Data (Bonferroni CI)

	Simultaneous_L	Simulatneous_U	Bonferroni_L	Bonferroni_U
d1	-0.5127078	0.6460411	-0.3744357	0.5077690
d2	-0.7078322	0.4411655	-0.5707236	0.3040570
d3	-0.7788034	0.1788034	-0.6645333	0.0645333
d4	-0.6289758	0.3623091	-0.5106869	0.2440202

• Example 7-11: Spouse Data (Question 2)

  Test Statistics

  Test	Value
  T^2	6.4266211
  F	1.4404495
  df1	4.0000000
  df2	26.0000000
  p	0.2488886

• Example 7-13: Swiss Banknotes (Two-Sample Hotelling’s)

Mean values for real and fake groups

Group	Length	Left	Right	Bottom	Top	Diag
Genuine	214.969	129.943	129.720	8.305	10.168	141.517
Counterfeit	214.823	130.300	130.193	10.530	11.133	139.450

Variance-covariance Matrix for Genuine Notes

	length	left	right	bottom	top	diag
length	0.1502414	0.0580131	0.0572929	0.0571263	0.0144525	0.0054818
left	0.0580131	0.1325768	0.0858990	0.0566515	0.0490667	-0.0430616
right	0.0572929	0.0858990	0.1262626	0.0581818	0.0306465	-0.0237778
bottom	0.0571263	0.0566515	0.0581818	0.4132071	-0.2634747	-0.0001869
top	0.0144525	0.0490667	0.0306465	-0.2634747	0.4211879	-0.0753091
diag	0.0054818	-0.0430616	-0.0237778	-0.0001869	-0.0753091	0.1998091

Variance-covariance Matrix for Counterfeit Notes

	length	left	right	bottom	top	diag
length	0.1240111	0.0315152	0.0240010	-0.1005960	0.0194354	0.0115657
left	0.0315152	0.0650505	0.0467677	-0.0240404	-0.0119192	-0.0050505
right	0.0240010	0.0467677	0.0889404	-0.0185758	0.0001323	0.0341919
bottom	-0.1005960	-0.0240404	-0.0185758	1.2813131	-0.4901919	0.2384848
top	0.0194354	-0.0119192	0.0001323	-0.4901919	0.4044556	-0.0220707
diag	0.0115657	-0.0050505	0.0341919	0.2384848	-0.0220707	0.3112121

Pooled Variance-Covariance Matrix

	length	left	right	bottom	top	diag
length	0.1371263	0.0447641	0.0406470	-0.0217348	0.0169439	0.0085237
left	0.0447641	0.0988136	0.0663333	0.0163056	0.0185737	-0.0240561
right	0.0406470	0.0663333	0.1076015	0.0198030	0.0153894	0.0052071
bottom	-0.0217348	0.0163056	0.0198030	0.8472601	-0.3768333	0.1191490
top	0.0169439	0.0185737	0.0153894	-0.3768333	0.4128217	-0.0486899
diag	0.0085237	-0.0240561	0.0052071	0.1191490	-0.0486899	0.2555106

• Example 7-14: Women’s Health Survey

              [,1]
diff1  0.117944052
diff2  0.354730710
diff3 -0.047179834
diff4  0.002835103
            diff1       diff2       diff3       diff4
diff1  0.19164212 -0.07101776  0.04511467 -0.03756199
diff2 -0.07101776  0.16538367 -0.17884359  0.02955601
diff3  0.04511467 -0.17884359  4.12372604 -3.75507101
diff4 -0.03756199  0.02955601 -3.75507101  4.39690660

Statistic	Value
T^2	1030.795
F	256.648
df1	4.000
df2	733.000
p	0.000

• Example 7-15: Swiss Banknotes

Confidence Intervals - Swiss Bank Notes

variable	n1	xbar1	s21	n2	xbar2	s22	sp	losim	upsim	lobon	upbon
bottom	100	8.305	0.4132071	100	10.530	1.2813131	0.8472601	-2.6980943	-1.7519057	-2.5719164	-1.8780836
diag	100	141.517	0.1998091	100	139.450	0.3112121	0.2555106	1.8071972	2.3268028	1.8764886	2.2575114
left	100	129.943	0.1325768	100	130.300	0.0650505	0.0988136	-0.5185652	-0.1954348	-0.4754745	-0.2385255
length	100	214.969	0.1502414	100	214.823	0.1240111	0.1371263	-0.0443267	0.3363267	0.0064349	0.2855651
right	100	129.720	0.1262626	100	130.193	0.0889404	0.1076015	-0.6415965	-0.3044035	-0.5966305	-0.3493695
top	100	10.168	0.4211879	100	11.133	0.4044556	0.4128217	-1.2952331	-0.6347669	-1.2071574	-0.7228426

Simultaneous 95% Confidence Intervals - Swiss Bank Notes

variable	losim	upsim
bottom	-2.6980943	-1.7519057
diag	1.8071972	2.3268028
left	-0.5185652	-0.1954348
length	-0.0443267	0.3363267
right	-0.6415965	-0.3044035
top	-1.2952331	-0.6347669

• 7.2.4 - Bonferroni Corrected (1 - α) x 100% Confidence Intervals

  95% Simultaneous Confidence Intervals (Bonferroni corrected) - Swiss Bank Notes

  variable	losim_adj	upsim_adj
  bottom	-2.5719890	-1.8780110
  diag	1.8764488	2.2575512
  left	-0.4754993	-0.2385007
  length	0.0064057	0.2855943
  right	-0.5966564	-0.3493436
  top	-1.2072080	-0.7227920

• Profile Plots

• 7.2.6 - Example (Box’s Test )

Warning: package 'biotools' was built under R version 4.3.3

    Box's M-test for Homogeneity of Covariance Matrices

data:  swiss[, -1]
Chi-Sq (approx.) = 121.9, df = 21, p-value = 3.198e-16

• 7.2.7 Example

Statistic	Value
T^2	2412.451
F	391.922
df1	6.000
df2	193.000
p	0.000

• Example 7-16: Swiss Bank Notes

Simultaneous (1 - α) x 100% Confidence Intervals

variable	losim_adj	upsim_adj
bottom	-2.6868875	-1.7631125
diag	1.8133515	2.3206485
left	-0.5147380	-0.1992620
length	-0.0398182	0.3318182
right	-0.6376027	-0.3083973
top	-1.2874105	-0.6425895

Lesson 8: Multivariate Analysis of Variance (MANOVA)

• Example 8-1 Pottery Data (MANOVA)

  site       ral          rfe        rmg          rca         rna
1    L  1.835714  0.627857143 -0.5264286 -0.052142857  0.25928571
2    L  1.235714  0.707857143 -1.3964286 -0.082142857 -0.08071429
3    L  2.035714  0.717857143 -0.9464286 -0.072142857 -0.05071429
4    L -1.064286 -0.002142857  0.8135714 -0.042142857 -0.11071429
5    L  1.235714  0.687857143  0.5135714 -0.002142857 -0.05071429
6    L -1.664286 -0.112142857 -1.3564286 -0.032142857 -0.03071429

• Example 8-2: Pottery Data

[1] 0.7118787

Test statistic: -2.131628e-14 

Degrees of freedom: 45 

Critical value ( 0.95 ): 61.65623 

Fail to reject the null hypothesis: Covariance matrices are homogeneous.

• Example 8-3: Pottery Data (MANOVA)

          Df    Wilks approx F num Df den Df   Pr(>F)    
site       3 0.012301   13.088     15 50.091 1.84e-12 ***
Residuals 22                                             
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• Example 8-4: Pottery Data (ANOVA)

 Response 1 :
            Df  Sum Sq Mean Sq F value    Pr(>F)    
site         3 175.610  58.537  26.669 1.627e-07 ***
Residuals   22  48.288   2.195                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

 Response 2 :
            Df  Sum Sq Mean Sq F value    Pr(>F)    
site         3 134.222  44.741  89.883 1.679e-12 ***
Residuals   22  10.951   0.498                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

 Response 3 :
            Df Sum Sq Mean Sq F value    Pr(>F)    
site         3 103.35  34.450   49.12 6.452e-10 ***
Residuals   22  15.43   0.701                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

 Response 4 :
            Df   Sum Sq  Mean Sq F value    Pr(>F)    
site         3 0.204703 0.068234  29.157 7.546e-08 ***
Residuals   22 0.051486 0.002340                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

 Response 5 :
            Df  Sum Sq  Mean Sq F value    Pr(>F)    
site         3 0.25825 0.086082  9.5026 0.0003209 ***
Residuals   22 0.19929 0.009059                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• Example 8-8: Pottery Data

        C  L  A   I
C+L-A-I 8 -2  8 -14
A_vs_I  1  0 -1   0
C_vs_L  0  1  0  -1

Call:
lm(formula = outcome ~ site, data = pottery)

Coefficients:
             [,1]     [,2]     [,3]     [,4]     [,5]   
(Intercept)  17.3200   1.5120   0.6060   0.0520   0.0480
siteC        -5.6200   3.9030   3.2490   0.2430   0.0020
siteI         0.8600   0.2000   0.0680  -0.0260   0.0060
siteL        -4.7557   4.8601   4.2204   0.1501   0.2027