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

Country	Meat	2000	2001	2002	2003	2004	2005	2006
Brazil	Beef	6102	6191	6437	6273	6400	6774	6939
Brazil	Pork	1827	1919	1975	1957	1979	1949	2191
Brazil	Poultry	5110	5341	5873	5742	5992	6612	6853
China	Beef	5284	5434	5818	6274	6703	7026	7395
China	Pork	40378	41829	43238	45054	46648	49703	51809
China	Poultry	9393	9237	9556	9963	9931	10088	10371
EuropeanUnion	Beef	8106	7658	8187	8315	8292	8194	8270
EuropeanUnion	Pork	19242	19317	19746	20043	19773	19768	20015
EuropeanUnion	Poultry	6934	7359	7417	7312	7280	7596	7380
Japan	Beef	1585	1419	1319	1366	1182	1200	1173
Japan	Pork	2228	2268	2377	2373	2562	2507	2450
Japan	Poultry	1772	1797	1830	1841	1713	1880	1908
Mexico	Beef	2309	2341	2409	2308	2368	2419	2509
Mexico	Pork	1252	1298	1349	1423	1556	1556	1580
Mexico	Poultry	2163	2311	2424	2627	2713	2871	3005
Russia	Beef	2246	2400	2450	2378	2308	2503	2370
Russia	Pork	2019	2076	2453	2420	2337	2476	2637
Russia	Poultry	1320	1588	1697	1680	1675	2139	2382
UnitedStates	Beef	12502	12351	12737	12340	12667	12663	12830
UnitedStates	Pork	8455	8389	8685	8816	8817	8669	8640
UnitedStates	Poultry	11474	11558	12270	12540	13080	13430	13754

Exploratory data analyses (EDA)

In the following analysis, we will aim to perform an analysis of variance (ANOVA) to compare the meat consumption amounts between different countries and/or across time. Note that the data points for each country-meat type combination are from the various years. Typically, we would expect the amount not to change between the years (especially in this 7-year timespan). Even if it did, in assuming homoscedasticity, we are making the assumption that any increase or decrease is constant between countries. Applying the Fligner-Killeen test will help us decide if this assumption is valid. Look at the bar graphs listed below and note which of them seem to vary more than the others between the years.

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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