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)

ANOVA

Open the SOCR ANOVA-Two Way applet (requires Java-enabled browser). For the following analyses, we will focus on the data for Beef consumption. Let us test to see if different countries eat different amounts of beef. Usually when we compare a set of groups as this one, we would use a one-way ANOVA (comparing the seven countries). To run this test, open up the one-way ANOVA analysis in the SOCR Analyses Applet in a java-enabled browser. It should be the default setting when you open up the page:

Now, prepare your dataset (it will be the Beef table from the above summary tables). We will be treating the yearly results as being our sample’s data points, attempting to capture the overall population consumption. This seems reasonable; the average meet consumption of a country should not change that much in a seven-year timespan.

Once you have rearranged your dataset for use in the ANOVA applet (if you do not know what it should look like, try considering one of the SOCR ANOVA tutorials). It should look like this in the applet data screen:

Rename your column headers to define the independent and dependent variables.

Click on the mapping tab, and assign your independent and dependent variables appropriately:

Set your precision to all, then click calculate:

The following results should appear in the Results tab:

Sample Size = 49

Independent Variable = Country

Dependent Variable = Consumption

Results of One-Way Analysis of Variance:

Standard 1-Way ANOVA Table

VarianceSource	DF	RSS	MSS	F-Statistics	P-value
TreatmentEffect(B/w_Groups)	6	668292765.43	111382127.57	898.89	< 1E-15
Error	42	5204240.57	123910.48
Total	48	673497006.00

Model:

Degrees of Freedom = 6

Residual Sum of Squares = 668292765.43

Mean Square Error = 111382127.57

Error:

Degrees of Freedom = 42

Residual Sum of Squares = 5204240.57

Mean Square Error = 123910.49

Corrected Total:

Degrees of Freedom = 48

Residual Sum of Squares = 673497006.00

F-Value = 898.89

P-Value = < 1E-15

R-Square = 0.99

Fligner-Killeen Analysis

Note our extremely low p-value. The results of our one-way ANOVA reject the null hypothesis that these countries do not differ in terms of beef consumption. This shouldn’t surprise anyone. Even a third grader could have made that decision after looking at the summary table, or the earlier EDA graphs. Besides, the data represents total volume of meat consumption, whereas the populations of these countries are vastly different, and a per-capita meat consumption analysis may be more appropriate in this case.

However, let’s say that you find a book that claims, using the results of your ANOVA as proof, that “China consumes more beef than any country other than the United States”. That book is saying that these populations are significantly different in terms of their beef consumption. Is this claim justified by the ANOVA analysis above? Going through the assumptions of the ANOVA test, remember that the different levels must follow homoscedasticity (equivalence of variance). If two populations are significantly far from this assumption, then they should be compared using ANOVA. An alternative statistical test needs to be applied (e.g., Kruskal-Wallis test).

To test for homoscedacity, we will use the Fligner-Killeen test, which is a non-parametric test that doesn’t assume normality. This is important, considering that we have a rather limited data set (only seven points, one for each year) and the data may not be normally distributed. Use the tab to find the Fligner-Killeen analysis in the SOCR Analyses Applet in a java-enabled browser:

Next, copy and paste your data into the spreadsheet using the paste button built into the applet. The data copied here is from the Beef summary table (excluding the marginal means). Note that we are treating the cases within each country (one for each year) as the data points. When trends take many years to happen, we can assume that, within a short timespan (such as the 7 years captured here), each year serves as estimates of a generally unchanging population value for that country.

Re-name the data headers to the country names, our levels for this comparison.

Click on the mapping tap and add all of the country names to the next tab.

Finally, select the precision setting “All.” This will display the maximum number of decimal numbers. Afterwards, click “CALCULATE”:

The following results text should appear on the screen:

Group Brazil

median = 6400.0

Group China

median = 6274.0

Group Europe

median = 8194.0

Group Japan

median = 1319.0

Group Mexico

median = 2368.0

Group Russia

median = 2378.0

Group United States

median = 12663.0

Total Size = 49

Total Mean Score = .784

Total Variance = .323

Degrees of Freedom = 6

Chi-Square Statistic = 19.421

P-Value = .004

The result from the Fligner-Killeen test (for variance homogeneity) yields a rather small p-value (0.004). Assuming α = 0.05, we would rejects the null hypothesis of equal variances between countries. Therefore, we should be wary of continuing with a standard (parametric) Analysis of Variance (ANOVA). Looking back at the data, it is easy to see why.

Conclusions

Note, for example, that in the data for the Chinese consumption of beef, the values are steadily rising between the years. In our ANOVA, we had made an assumption, that the overall consumption would not follow a set pattern between the years, allowing us to treat them as sample points. This results in a much different variance between the groups. Therefore, it would be incorrect to use our previous conclusion from the ANOVA that “these countries have different population values for meat consumption” without a qualifier of the date range that we were sampling within. Note that without this analysis, you might have missed the trend in Chinese beef consumption—which might be worth studying in its own right.

We can also think about this from a more mathematical perspective. This brings up the fundamental definition of Analysis of Variance, in which we are trying to keep constant the within-group variance of each of our levels (in this case, countries). If we ignore homoscedasticity, from the viewpoint of the algebra involved, we are essentially comparing completely unrelated populations.

Just because the assumptions of ANOVA are not satisfied, we should not give up on this data set. Without the results of the Fligner-Killeen, we may have missed the effects of time on meat consumption, an effect that warrants is own investigation. In such situations, the use of non-parametric alternatives to ANOVA may be appropriate (e.g., Kruskal-Wallis test).

Practice problems

• Run this same analysis for the pork and poultry datasets. Interpret any conclusions as shown above.
• Practice running a one-way ANOVA, excluding the data from China and Japan.
• “Turn” the beef data set around, and run this Fligner-Killeen test between the year groups. Interpret the conclusion you find. Why is this same/different to the result from the above example?

See also

• Fligner-Killeen activity
• Kruskal-Wallis activity
• NOVA activity and EBook Chapter

References

• Meat Consumption by Type and Country (2006). U.S. Department of Agriculture, Foreign Agricultural Service, Livestock, and Poultry.

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