Beispieldaten Korrelation in VL

Statistik: Übung

Author

Affiliation

Prof. Dr. Armin Eichinger

TH Deggendorf

Aufgabe

Gegeben sind die folgenden Daten zu Größe und Gewicht, die im Rahmen einer Vorlesung erhoben worden sind:

  height weight
1    191     81
2    172     56
3    190    110
4    165     55
5    177     86
6    167     65

Bestimmen Sie mit Hilfe von grundlegenden R-Funktionen (alternativ: Taschenrechner):

• Kovarianz
• Standardabweichungen
• Korrelationskoeffizient r
• 95%-Kondidenzintervall für r
• Signifikanztest für r
• p-Wert (geht mit Taschenrechner nicht)

Lösung ab hier

Größe und Gewicht: Daten einlesen und r erzeugen

Vorarbeit: MW, SD, COV

grgw_data <- data.frame(height=c(191,172, 190, 165, 177, 167), 
                        weight=c(81, 56, 110, 55, 86, 65))

# Daten transformiert in Ränge
#grgw_data <- data.frame(height=c(1, 4, 2, 6, 3, 5), 
#                        weight=c(3, 5, 1, 6, 2, 4))

head(grgw_data)

  height weight
1    191     81
2    172     56
3    190    110
4    165     55
5    177     86
6    167     65

mw_w <- mean(grgw_data$weight)
mw_h <- mean(grgw_data$height)

mw_h

[1] 177

mw_w

[1] 75.5

# Kovarianz
my_cov <- cov(grgw_data$height, grgw_data$weight)
my_cov

[1] 194.8

# Standardabweichungen
sd_w <- sd(grgw_data$weight)
sd_h <- sd(grgw_data$height)

sd_h

[1] 11.26055

sd_w

[1] 21.1731

# r händisch
my_r <- my_cov/(sd_w*sd_h)
my_r

[1] 0.8170431

# r mit Funtion cor()
my_r_2 <- cor(grgw_data$height,grgw_data$weight)
my_r_2

[1] 0.8170431

Konfidenzintervall

# Stichprobengröße n
my_n <- length(grgw_data$height )

# Standardfehler
my_se <- 1/sqrt(my_n - 3)

# Fisher-Z-Transformation r → z
my_r_fz <- 0.5*log((1+my_r)/(1-my_r))

# oberes und unteres Ende des KI
ki_down_z <- my_r_fz - 1.96*my_se
ki_up_z <- my_r_fz + 1.96*my_se 

ki_down_z

[1] 0.01625101

ki_up_z

[1] 2.279464

# oberes und unteres Ende des KI rücktransformieren (z→r)
ki_down <- (exp(2*ki_down_z)-1)/(exp(2*ki_down_z)+1)
ki_up <- (exp(2*ki_up_z)-1)/(exp(2*ki_up_z)+1)

ki_down

[1] 0.01624957

ki_up

[1] 0.9792705

Signifikanztest

Annahme: Gerichtete Hypothese (positive Korrelation)

# Teststatistik t berechnen (= empirischer t-Wert)
my_t <- (my_r * sqrt(my_n-2)) / sqrt(1-my_r*my_r)
my_t

[1] 2.834118

# kritischer t-Wert
my_t_krit <- qt(0.95,my_n-2)
my_t_krit

[1] 2.131847

# p-Wert
my_p <- pt(my_t,length(grgw_data$height) - 2,lower.tail = FALSE)
my_p

[1] 0.02357389

Anhang: t-Verteilung

df	0.01	0.025	0.05	0.10	0.25	0.5	0.75	0.90	0.95	0.975	0.99
1	-31.8205	-12.7062	-6.3138	-3.0777	-1.0000	0	1.0000	3.0777	6.3138	12.7062	31.8205
2	-6.9646	-4.3027	-2.9200	-1.8856	-0.8165	0	0.8165	1.8856	2.9200	4.3027	6.9646
3	-4.5407	-3.1824	-2.3534	-1.6377	-0.7649	0	0.7649	1.6377	2.3534	3.1824	4.5407
4	-3.7469	-2.7764	-2.1318	-1.5332	-0.7407	0	0.7407	1.5332	2.1318	2.7764	3.7469
5	-3.3649	-2.5706	-2.0150	-1.4759	-0.7267	0	0.7267	1.4759	2.0150	2.5706	3.3649
6	-3.1427	-2.4469	-1.9432	-1.4398	-0.7176	0	0.7176	1.4398	1.9432	2.4469	3.1427
7	-2.9980	-2.3646	-1.8946	-1.4149	-0.7111	0	0.7111	1.4149	1.8946	2.3646	2.9980
8	-2.8965	-2.3060	-1.8595	-1.3968	-0.7064	0	0.7064	1.3968	1.8595	2.3060	2.8965
9	-2.8214	-2.2622	-1.8331	-1.3830	-0.7027	0	0.7027	1.3830	1.8331	2.2622	2.8214
10	-2.7638	-2.2281	-1.8125	-1.3722	-0.6998	0	0.6998	1.3722	1.8125	2.2281	2.7638
11	-2.7181	-2.2010	-1.7959	-1.3634	-0.6974	0	0.6974	1.3634	1.7959	2.2010	2.7181
12	-2.6810	-2.1788	-1.7823	-1.3562	-0.6955	0	0.6955	1.3562	1.7823	2.1788	2.6810
13	-2.6503	-2.1604	-1.7709	-1.3502	-0.6938	0	0.6938	1.3502	1.7709	2.1604	2.6503
14	-2.6245	-2.1448	-1.7613	-1.3450	-0.6924	0	0.6924	1.3450	1.7613	2.1448	2.6245
15	-2.6025	-2.1314	-1.7531	-1.3406	-0.6912	0	0.6912	1.3406	1.7531	2.1314	2.6025
16	-2.5835	-2.1199	-1.7459	-1.3368	-0.6901	0	0.6901	1.3368	1.7459	2.1199	2.5835
17	-2.5669	-2.1098	-1.7396	-1.3334	-0.6892	0	0.6892	1.3334	1.7396	2.1098	2.5669
18	-2.5524	-2.1009	-1.7341	-1.3304	-0.6884	0	0.6884	1.3304	1.7341	2.1009	2.5524
19	-2.5395	-2.0930	-1.7291	-1.3277	-0.6876	0	0.6876	1.3277	1.7291	2.0930	2.5395
20	-2.5280	-2.0860	-1.7247	-1.3253	-0.6870	0	0.6870	1.3253	1.7247	2.0860	2.5280
21	-2.5176	-2.0796	-1.7207	-1.3232	-0.6864	0	0.6864	1.3232	1.7207	2.0796	2.5176
22	-2.5083	-2.0739	-1.7171	-1.3212	-0.6858	0	0.6858	1.3212	1.7171	2.0739	2.5083
23	-2.4999	-2.0687	-1.7139	-1.3195	-0.6853	0	0.6853	1.3195	1.7139	2.0687	2.4999
24	-2.4922	-2.0639	-1.7109	-1.3178	-0.6848	0	0.6848	1.3178	1.7109	2.0639	2.4922
25	-2.4851	-2.0595	-1.7081	-1.3163	-0.6844	0	0.6844	1.3163	1.7081	2.0595	2.4851
26	-2.4786	-2.0555	-1.7056	-1.3150	-0.6840	0	0.6840	1.3150	1.7056	2.0555	2.4786
27	-2.4727	-2.0518	-1.7033	-1.3137	-0.6837	0	0.6837	1.3137	1.7033	2.0518	2.4727
28	-2.4671	-2.0484	-1.7011	-1.3125	-0.6834	0	0.6834	1.3125	1.7011	2.0484	2.4671
29	-2.4620	-2.0452	-1.6991	-1.3114	-0.6830	0	0.6830	1.3114	1.6991	2.0452	2.4620
30	-2.4573	-2.0423	-1.6973	-1.3104	-0.6828	0	0.6828	1.3104	1.6973	2.0423	2.4573
31	-2.4528	-2.0395	-1.6955	-1.3095	-0.6825	0	0.6825	1.3095	1.6955	2.0395	2.4528
32	-2.4487	-2.0369	-1.6939	-1.3086	-0.6822	0	0.6822	1.3086	1.6939	2.0369	2.4487
33	-2.4448	-2.0345	-1.6924	-1.3077	-0.6820	0	0.6820	1.3077	1.6924	2.0345	2.4448
34	-2.4411	-2.0322	-1.6909	-1.3070	-0.6818	0	0.6818	1.3070	1.6909	2.0322	2.4411
35	-2.4377	-2.0301	-1.6896	-1.3062	-0.6816	0	0.6816	1.3062	1.6896	2.0301	2.4377
36	-2.4345	-2.0281	-1.6883	-1.3055	-0.6814	0	0.6814	1.3055	1.6883	2.0281	2.4345
37	-2.4314	-2.0262	-1.6871	-1.3049	-0.6812	0	0.6812	1.3049	1.6871	2.0262	2.4314
38	-2.4286	-2.0244	-1.6860	-1.3042	-0.6810	0	0.6810	1.3042	1.6860	2.0244	2.4286
39	-2.4258	-2.0227	-1.6849	-1.3036	-0.6808	0	0.6808	1.3036	1.6849	2.0227	2.4258
40	-2.4233	-2.0211	-1.6839	-1.3031	-0.6807	0	0.6807	1.3031	1.6839	2.0211	2.4233
41	-2.4208	-2.0195	-1.6829	-1.3025	-0.6805	0	0.6805	1.3025	1.6829	2.0195	2.4208
42	-2.4185	-2.0181	-1.6820	-1.3020	-0.6804	0	0.6804	1.3020	1.6820	2.0181	2.4185
43	-2.4163	-2.0167	-1.6811	-1.3016	-0.6802	0	0.6802	1.3016	1.6811	2.0167	2.4163
44	-2.4141	-2.0154	-1.6802	-1.3011	-0.6801	0	0.6801	1.3011	1.6802	2.0154	2.4141
45	-2.4121	-2.0141	-1.6794	-1.3006	-0.6800	0	0.6800	1.3006	1.6794	2.0141	2.4121
46	-2.4102	-2.0129	-1.6787	-1.3002	-0.6799	0	0.6799	1.3002	1.6787	2.0129	2.4102
47	-2.4083	-2.0117	-1.6779	-1.2998	-0.6797	0	0.6797	1.2998	1.6779	2.0117	2.4083
48	-2.4066	-2.0106	-1.6772	-1.2994	-0.6796	0	0.6796	1.2994	1.6772	2.0106	2.4066
49	-2.4049	-2.0096	-1.6766	-1.2991	-0.6795	0	0.6795	1.2991	1.6766	2.0096	2.4049
50	-2.4033	-2.0086	-1.6759	-1.2987	-0.6794	0	0.6794	1.2987	1.6759	2.0086	2.4033
51	-2.4017	-2.0076	-1.6753	-1.2984	-0.6793	0	0.6793	1.2984	1.6753	2.0076	2.4017
52	-2.4002	-2.0066	-1.6747	-1.2980	-0.6792	0	0.6792	1.2980	1.6747	2.0066	2.4002
53	-2.3988	-2.0057	-1.6741	-1.2977	-0.6791	0	0.6791	1.2977	1.6741	2.0057	2.3988
54	-2.3974	-2.0049	-1.6736	-1.2974	-0.6791	0	0.6791	1.2974	1.6736	2.0049	2.3974
55	-2.3961	-2.0040	-1.6730	-1.2971	-0.6790	0	0.6790	1.2971	1.6730	2.0040	2.3961
56	-2.3948	-2.0032	-1.6725	-1.2969	-0.6789	0	0.6789	1.2969	1.6725	2.0032	2.3948
57	-2.3936	-2.0025	-1.6720	-1.2966	-0.6788	0	0.6788	1.2966	1.6720	2.0025	2.3936
58	-2.3924	-2.0017	-1.6716	-1.2963	-0.6787	0	0.6787	1.2963	1.6716	2.0017	2.3924
59	-2.3912	-2.0010	-1.6711	-1.2961	-0.6787	0	0.6787	1.2961	1.6711	2.0010	2.3912
60	-2.3901	-2.0003	-1.6706	-1.2958	-0.6786	0	0.6786	1.2958	1.6706	2.0003	2.3901
61	-2.3890	-1.9996	-1.6702	-1.2956	-0.6785	0	0.6785	1.2956	1.6702	1.9996	2.3890
62	-2.3880	-1.9990	-1.6698	-1.2954	-0.6785	0	0.6785	1.2954	1.6698	1.9990	2.3880
63	-2.3870	-1.9983	-1.6694	-1.2951	-0.6784	0	0.6784	1.2951	1.6694	1.9983	2.3870
64	-2.3860	-1.9977	-1.6690	-1.2949	-0.6783	0	0.6783	1.2949	1.6690	1.9977	2.3860
65	-2.3851	-1.9971	-1.6686	-1.2947	-0.6783	0	0.6783	1.2947	1.6686	1.9971	2.3851
66	-2.3842	-1.9966	-1.6683	-1.2945	-0.6782	0	0.6782	1.2945	1.6683	1.9966	2.3842
67	-2.3833	-1.9960	-1.6679	-1.2943	-0.6782	0	0.6782	1.2943	1.6679	1.9960	2.3833
68	-2.3824	-1.9955	-1.6676	-1.2941	-0.6781	0	0.6781	1.2941	1.6676	1.9955	2.3824
69	-2.3816	-1.9949	-1.6672	-1.2939	-0.6781	0	0.6781	1.2939	1.6672	1.9949	2.3816
70	-2.3808	-1.9944	-1.6669	-1.2938	-0.6780	0	0.6780	1.2938	1.6669	1.9944	2.3808
71	-2.3800	-1.9939	-1.6666	-1.2936	-0.6780	0	0.6780	1.2936	1.6666	1.9939	2.3800
72	-2.3793	-1.9935	-1.6663	-1.2934	-0.6779	0	0.6779	1.2934	1.6663	1.9935	2.3793
73	-2.3785	-1.9930	-1.6660	-1.2933	-0.6779	0	0.6779	1.2933	1.6660	1.9930	2.3785
74	-2.3778	-1.9925	-1.6657	-1.2931	-0.6778	0	0.6778	1.2931	1.6657	1.9925	2.3778
75	-2.3771	-1.9921	-1.6654	-1.2929	-0.6778	0	0.6778	1.2929	1.6654	1.9921	2.3771
76	-2.3764	-1.9917	-1.6652	-1.2928	-0.6777	0	0.6777	1.2928	1.6652	1.9917	2.3764
77	-2.3758	-1.9913	-1.6649	-1.2926	-0.6777	0	0.6777	1.2926	1.6649	1.9913	2.3758
78	-2.3751	-1.9908	-1.6646	-1.2925	-0.6776	0	0.6776	1.2925	1.6646	1.9908	2.3751
79	-2.3745	-1.9905	-1.6644	-1.2924	-0.6776	0	0.6776	1.2924	1.6644	1.9905	2.3745
80	-2.3739	-1.9901	-1.6641	-1.2922	-0.6776	0	0.6776	1.2922	1.6641	1.9901	2.3739
81	-2.3733	-1.9897	-1.6639	-1.2921	-0.6775	0	0.6775	1.2921	1.6639	1.9897	2.3733
82	-2.3727	-1.9893	-1.6636	-1.2920	-0.6775	0	0.6775	1.2920	1.6636	1.9893	2.3727
83	-2.3721	-1.9890	-1.6634	-1.2918	-0.6775	0	0.6775	1.2918	1.6634	1.9890	2.3721
84	-2.3716	-1.9886	-1.6632	-1.2917	-0.6774	0	0.6774	1.2917	1.6632	1.9886	2.3716
85	-2.3710	-1.9883	-1.6630	-1.2916	-0.6774	0	0.6774	1.2916	1.6630	1.9883	2.3710
86	-2.3705	-1.9879	-1.6628	-1.2915	-0.6774	0	0.6774	1.2915	1.6628	1.9879	2.3705
87	-2.3700	-1.9876	-1.6626	-1.2914	-0.6773	0	0.6773	1.2914	1.6626	1.9876	2.3700
88	-2.3695	-1.9873	-1.6624	-1.2912	-0.6773	0	0.6773	1.2912	1.6624	1.9873	2.3695
89	-2.3690	-1.9870	-1.6622	-1.2911	-0.6773	0	0.6773	1.2911	1.6622	1.9870	2.3690
90	-2.3685	-1.9867	-1.6620	-1.2910	-0.6772	0	0.6772	1.2910	1.6620	1.9867	2.3685
91	-2.3680	-1.9864	-1.6618	-1.2909	-0.6772	0	0.6772	1.2909	1.6618	1.9864	2.3680
92	-2.3676	-1.9861	-1.6616	-1.2908	-0.6772	0	0.6772	1.2908	1.6616	1.9861	2.3676
93	-2.3671	-1.9858	-1.6614	-1.2907	-0.6771	0	0.6771	1.2907	1.6614	1.9858	2.3671
94	-2.3667	-1.9855	-1.6612	-1.2906	-0.6771	0	0.6771	1.2906	1.6612	1.9855	2.3667
95	-2.3662	-1.9853	-1.6611	-1.2905	-0.6771	0	0.6771	1.2905	1.6611	1.9853	2.3662
96	-2.3658	-1.9850	-1.6609	-1.2904	-0.6771	0	0.6771	1.2904	1.6609	1.9850	2.3658
97	-2.3654	-1.9847	-1.6607	-1.2903	-0.6770	0	0.6770	1.2903	1.6607	1.9847	2.3654
98	-2.3650	-1.9845	-1.6606	-1.2902	-0.6770	0	0.6770	1.2902	1.6606	1.9845	2.3650
99	-2.3646	-1.9842	-1.6604	-1.2902	-0.6770	0	0.6770	1.2902	1.6604	1.9842	2.3646
100	-2.3642	-1.9840	-1.6602	-1.2901	-0.6770	0	0.6770	1.2901	1.6602	1.9840	2.3642