SOCR Activity ANOVA FlignerKilleen MeatConsumption

Motivation and Goals

In many developed countries, when people imagine their next meal, they focus on one specific part: the meat. That choice of meat, however, varies from country to country due to the popularity and availability of various domesticated animals. Furthermore, the amount of meat eaten has a surprising degree of variability across time, cultures and geographic regions.

The following activity will study the effects of that variance on the statistical analyses. Specifically, we will consider how deviations from homoscedasticity (also known as equivalence of variance or variance homogeneity) can lead to making some incomplete or even incorrect conclusions. To do so, we will employ the Fligner-Killeen method to analyze some real meet consumption data.

Summary

This activity uses a reduced version of the open-source meat-consumption dataset. All data comes from the US Census Bureau.

This dataset summarizes the meat consumption, by animal type, of various countries (the European Union (EU) is being treated as a single country in this case). For simplicity, records from countries that did provide consumption measures for all meat types and all years were removed from the data set.

Data

Data Description

• Number of cases: 147
• Variables
  • Country: The country or world region in question
    • Brazil
    • China
    • European Union
    • Japan
    • Mexico
    • Russia
    • United States
  • Meat: The type of meat
    • Beef
    • Pork
    • Poultry
  • Years Represented (2000 – 2006)
• Values are in thousands of metric tons

Data Summaries

Chicken/Poultry

Year	Brazil	China	Europe	Japan	Mexico	Russia	UnitedStates	YearAverage	YearSD
2000	5110	9393	6934	1772	2163	1320	11474	5452.286	3990.459
2001	5341	9237	7359	1797	2311	1588	11558	5598.714	3942.57
2002	5873	9556	7417	1830	2424	1697	12270	5866.714	4134.211
2003	5742	9963	7312	1841	2627	1680	12540	5957.857	4234.565
2004	5992	9931	7280	1713	2713	1675	13080	6054.857	4379.591
2005	6612	10088	7596	1880	2871	2139	13430	6373.714	4388.111
2006	6853	10371	7380	1908	3005	2382	13754	6521.857	4448.974
Country_Average	5931.857	9791.286	7325.429	1820.143	2587.714	1783	12586.57
Country_SD	629.6543	407.0908	200.4826	66.03895	304.2404	357.6777	886.5564

Pork

Year	Brazil	China	Europe	Japan	Mexico	Russia	UnitedStates	YearAverage	YearSD
2000	1827	40378	19242	2228	1252	2019	8455	10771.57	14570.99
2001	1919	41829	19317	2268	1298	2076	8389	11013.71	15049.33
2002	1975	43238	19746	2377	1349	2453	8685	11403.29	15502.7
2003	1957	45054	20043	2373	1423	2420	8816	11726.57	16145.49
2004	1979	46648	19773	2562	1556	2337	8817	11953.14	16648.16
2005	1949	49703	19768	2507	1556	2476	8669	12375.43	17714.83
2006	2191	51809	20015	2450	1580	2637	8640	12760.29	18438.64
Country_Average	1971	45522.71	19700.57	2395	1430.571	2345.429	8638.714
Country_SD	110	4159.521	312.355	121.3013	135.3808	223.0148	164.5121

Beef

Year	Brazil	China	Europe	Japan	Mexico	Russia	UnitedStates	YearAverage	YearSD
2000	6102	5284	8106	1585	2309	2246	12502	5447.714	3922.316
2001	6191	5434	7658	1419	2341	2400	12351	5399.143	3835.093
2002	6437	5818	8187	1319	2409	2450	12737	5622.429	4016.753
2003	6273	6274	8315	1366	2308	2378	12340	5607.714	3933.847
2004	6400	6703	8292	1182	2368	2308	12667	5702.857	4077.861
2005	6774	7026	8194	1200	2419	2503	12663	5825.571	4056.693
2006	6939	7395	8270	1173	2509	2370	12830	5926.571	4148.408
Country_Average	6445.143	6276.286	8146	1320.571	2380.429	2379.286	12584.29
Country_SD	307.1685	806.2036	226.9295	151.2138	71.75388	85.31259	190.4396

Raw Dataset

Sex	locality	shell.h	shell.w	aperture.h	aperture.w	whorl.w	rib.n
1	1	7063	3860	2564	2522	3119	29
1	1	7535	3846	2522	2629	3332	37
1	1	7484	3952	2680	2541	3258	34
1	1	7516	3763	2671	2703	3217	31
1	1	7211	3698	2555	2541	3133	30
1	1	7526	3828	2573	2555	3143	28
1	1	7576	3851	2439	2504	3207	32
1	1	7558	3781	2471	2481	3143	35
1	1	7674	3828	2652	2513	3230	31
1	1	7526	3786	2439	2555	3166	29
1	1	7641	3943	2587	2481	3300	30
1	1	7470	4087	2712	2606	3286	33
1	1	7262	3693	2573	2504	3124	38
1	1	7410	3781	2532	2499	3152	28
1	1	7799	4045	2782	2666	3342	31
1	1	7567	3855	2583	2647	3226	33
1	1	7428	3952	2481	2546	3226	32
1	1	7368	3730	2448	2407	3119	33
1	1	6915	3615	2407	2259	3069	29
1	1	7327	3920	2587	2522	3217	36
1	1	7502	3837	2657	2536	3087	32
2	1	7188	4003	2629	2546	3277	31
2	1	7178	4119	2754	2670	3258	25
2	1	7192	3865	2587	2374	3101	27
2	1	7035	3971	2652	2647	3207	31
2	1	6674	3823	2532	2573	3004	30
2	1	7470	4027	2837	2694	3184	27
2	1	7252	3952	2629	2407	3166	27
2	1	6739	3855	2606	2439	3087	29
2	1	7345	3994	2721	2555	3166	31
2	1	7419	3869	2689	2564	3198	29
2	1	7040	3837	2513	2481	3124	33
2	1	6623	3920	2564	2564	3124	26
2	1	7169	3980	2828	2434	3161	30
2	1	6956	3777	2615	2495	3050	29
2	1	6549	3749	2471	2425	2994	25
2	1	6831	3763	2536	2407	3036	30
2	1	7053	3985	2805	2573	3184	32
2	1	6919	3837	2481	2448	3152	32
2	1	6808	3911	2666	2499	3101	33
2	1	7137	3573	2471	2522	3050	27
2	1	6928	3851	2587	2495	3166	28
1	2	7294	3679	2458	2356	3101	33
1	2	7576	3786	2721	2777	3143	31
1	2	7840	3892	2712	2471	3082	32
1	2	7586	3860	2573	2647	3207	31
1	2	7715	3860	2536	2546	3184	33
1	2	7715	4017	2777	2638	3305	35
1	2	7456	3929	2629	2680	3203	32
1	2	7526	3878	2583	2606	3291	35
1	2	7526	3772	2606	2564	3152	33
1	2	7780	3772	2735	2629	3217	31
1	2	7077	3703	2305	2337	3092	34
1	2	7336	3781	2555	2439	3143	36
1	2	7544	3855	2583	2638	3119	36
2	2	7086	4077	2772	2629	3143	31
2	2	6716	3846	2694	2689	3069	29
2	2	7276	3929	2587	2481	3170	32
2	2	6928	3860	2712	2564	3082	34
2	2	6716	3818	2689	2439	3082	34
2	2	6915	3869	2721	2388	3069	35
2	2	7095	3929	2712	2490	3110	32
2	2	6928	3952	2754	2638	3092	30
2	2	7387	3804	2592	2564	3110	31
2	2	7095	3869	2615	2541	3272	33
2	2	7095	3846	2638	2587	3119	28
2	2	6938	3846	2638	2638	3069	32
2	2	7169	3721	2745	2629	3092	29
2	2	7035	3689	2513	2397	3018	29
2	2	7086	3786	2596	2356	3101	31
2	2	7123	4059	2731	2819	3217	34
2	2	7137	3878	2684	2546	3133	35
2	2	6956	3619	2407	2296	3082	31
2	2	7123	3929	2550	2541	3124	34
2	2	7220	3869	2657	2448	3207	33
2	2	7493	3971	2680	2638	3235	30
2	2	7271	3994	2638	2620	3193	33
1	3	7831	3948	2647	2708	3360	33
1	3	7863	4207	2712	2694	3416	30
1	3	7775	3952	2721	2573	3291	32
1	3	7715	4068	2689	2430	3300	29
1	3	7493	3892	2731	2578	3175	33
1	3	7243	4008	2689	2786	3110	32
1	3	8187	4207	2953	2902	3490	33
1	3	7850	4119	2721	2680	3499	35
1	3	8086	4128	2819	2680	3291	34
1	3	8211	3906	2647	2541	3277	31
1	3	7799	3929	2564	2499	3351	30
1	3	7683	3962	2721	2666	3272	31
1	3	7428	3952	2541	2527	3258	34
1	3	7715	3994	2754	2657	3291	36
1	3	7780	3994	2647	2680	3383	33
1	3	7567	3869	2583	2365	3249	34
1	3	7891	3943	2573	2564	3207	26
1	3	7526	3804	2638	2522	3217	32
1	3	7664	3888	2555	2541	3249	33
2	3	7003	3962	2564	2439	3268	34
2	3	7368	4077	2860	2712	3156	36
2	3	7276	3837	2504	2300	3143	30
2	3	7276	3814	2546	2374	3161	32
2	3	6989	3633	2189	2161	3277	35
2	3	7104	3814	2763	2680	3087	36
2	3	7123	3971	2721	2564	3217	32
2	3	7229	3855	2439	2388	3281	31
2	3	7396	4119	2596	2661	3268	30
2	3	7660	3721	2680	2458	3198	36
2	3	7044	3892	2735	2546	3036	34
2	3	7220	3772	2647	2416	3087	29
2	3	7317	4119	2911	2869	3235	32
2	3	6841	3962	2763	2703	3152	35
2	3	7294	4054	2754	2416	3166	32
2	3	7873	4193	2920	2893	3342	30

Exploratory data analyses (EDA)

Various data patterns may be observed and explored using different types of graphical tools for plotting variables. Which of the following graphs are more or less likely to demonstrate visually significant grouping differences?

Quantitative data analysis (QDA)

Open the SOCR ANOVA-Two Way applet (requires Java-enabled browser).

Copy and paste the Sex and Locality data into the first two columns. Pick one of the other six variables (in this case, Shell.h) and copy that data into the third. Use the ctrl + c command and the "paste" button in the applet. Name the three columns appropriately.

Next, click on the “mapping tab”. Select "sex" and "locality" as the independent variables. Next, name the third column as your dependent variable. We will use "shell.h" in the following example, but it is recommended that you use another in its place to explore these measures. Make sure you click “turn the interaction” on,

Press the Calculate button. This should bring up the results page with the following text:

ANOVA results

Sample Size = 112

Dependent Variable = Shell.h

Independent Variable(s) = Locality Sex Interaction Locality: Sex

*** Two-Way Analysis of Variance Results ***

See EBook's Standard 2-Way ANOVA Table

Variance Source	DF	RSS	MSS	F-Statistics	P-value
Main Effect: Locality	2	1912452.01667	956226.00833	18.39651	0.00000
Main Effect: Sex	1	6197835.01312	6197835.01312	119.23809	0.00000
Interaction Locality: Sex	2	161192.25392	80596.12696	1.55056	0.21690
Error	106	5509737.01359	51978.65107
Total:	111	13170123.10714

Variable: Locality

Degrees of Freedom = 2

Residual Sum of Squares = 1912452.01667

Mean Square Error = 956226.00833

F-Value = 18.39651

P-Value = .00000

Variable: Sex

Degrees of Freedom = 1

Residual Sum of Squares = 6197835.01312

Mean Square Error = 6197835.01312

F-Value = 119.23809

P-Value = .00000

Variable: Interaction Locality: Sex

Degrees of Freedom = 2

Residual Sum of Squares = 161192.25392

Mean Square Error = 80596.12696

F-Value = 1.55056

P-Value = .21690

Residual: Degrees of Freedom = 106

Residual Sum of Squares = 5509737.01359

Mean Square Error = 51978.65107

F-Value = 29.47512

P-Value = 0.0

R-Square = .60598

For the effect of locality and the interaction effects, you can need to conduct post-hoc t-tests, in this case, a pooled independent samples t-test. You can do this in a similar manner to the two-way ANOVA; however will have to enter the values in a slightly different way (see below). Note that your critical t-values must have Bonferoni correction.

Conclusions

According to the results of the analysis, you will find that there is are significant main effects of locality (F(2, 106) = 18.39651, p < 0.001) and sex (F(1, 106) = 119.23809, p < 0.001) on shell width. The interaction between sex and locality is not significant on shell width (F (2,106) = 1.55056, p > 0.20). Post-hoc tests reveal that t-tests will reveal that there is a significant difference in width between male (M 7106.88136, SD = 247.06778) and female (M = 7578.03773, SD = 256.89806) snails shells (t (110) = 9.88846, p < 0.001). The 99.7% confidence interval for the difference is 471.15638 ± 157.08993. Note that this interval does not include 0 (a lack of difference between the means). There is also a significant difference in width between the snails collected at localities one and two, two and three, & one and three. We leave these analyses to you in the first practice problems

Based on these results, it would be possible to classify whether a Cocholotoma septemspirale is male or female, regardless of the locality it comes from (there is no interaction of the two effects); females have significantly taller shells. Limitations of the study include its correlational nature. One issue with the study, for example, is that age might be a confounding variable, if these snails are the type that grows throughout their lifecycle.

Practice problems

• Finish the post-hoc t-tests for the effect of locality on shell width.
• Complete an analysis similar to the one above, using one of the variables other than shell.h as -your dependent variable. See if that variable would be of use in classifying the snails.
• Complete a new analysis of this pain/neuroimaging data set. Use sex and disease group as independent variables. Choose for your dependent variable one of the brain volumes.

See also

• SOCR 2-Way ANOVA Wiki Page
• Baseball Salary/Wins Activity

References

• Che, Annie, Cui, Jenny, and Dinov, Ivo (2009). SOCR Analyses: Implementation and Demonstration of a New Graphical Statistics Educational Toolkit. JSS, Vol. 30, Issue 3, Apr 2009.
• Che, A, Cui, J, and Dinov, ID (2009) SOCR Analyses – an Instructional Java Web-based Statistical Analysis Toolkit, JOLT, 5(1), 1-19, March 2009.
• Dinov, ID. Statistics Online Computational Resource, Journal of Statistical Software, Vol. 16, No. 1, 1-16, October 2006.
• Reichenbach F, Baur H, Neubert E (2012) Sexual dimorphism in shells of Cochlostoma septemspirale (Caenogastropoda, Cyclophoroidea, Diplommatinidae, Cochlostomatinae). ZooKeys 208: 1-16. doi:10.3897/zookeys.208.2869
• Baur H, Reichenbach F, Neubert E (2012) Data from: Sexual dimorphism in shells of Cochlostoma septemspirale (Caenogastropoda, Cyclophoroidea, Diplommatinidae, Cochlostomatinae). Dryad Digital Repository. doi:10.5061/dryad.ns7v7

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