Chapter 6: Tests of Independence



Tests of Independence

Purpose:

Tests of independence are used to determine two or more discrete variables are potentially related to each other or independent of each other.

1 Background:

We will be examining three versions of tests of independence, 2X2 Test of Independence, RXC Test of Independence and Log-linear analysis. The latter of which will be examined separately in another lab exercise. The tests differ in the number of variables or number of levels (i.e. possible values) of the variables that can be used (Table 5-1).

Table 5- 1: Which Test of Independence to use.

|# of Variables |# of Levels per Variable |Test |

|2 |2 |2X2 |

|2 |2 for one variable and more than 2 for the other |RXC |

|2 or more |Any variable can have more than 2 levels |Log-linear |

The tests essentially compare the observed frequencies expressed as proportions of the levels of one variable with respect to the observed proportions of the levels of the other variable. If the proportions between levels of one variable are essentially the same for both levels of the other variable, then one concludes that the two variables are not related or are independent of each other.

For example, let’s assume that we are trying to see if a pesticide is effective in reducing the number of pests. We treated 40 plants with pesticide and left 40 plants untreated. Then, several days later, we examined the plants to see how many were infested and how many were not. The two variables are Pesticide (with levels: Yes and No) and Infested (with levels: Yes and No). Let’s assume that, for either level of Pesticide, 28% (expressed as a proportion =0.28) of the plants became infested and 72% did not (Table 5-1: Case 1). In this case, whether or not a plant becomes infested is INDEPENDENT of whether or not the pesticide was applied. In another situation (Table 5-1: Case 2) assume that, when there is no pesticide, 85% of the plants are infested but, when the pesticide is used, only 25% of the plants are infested. In this case, whether or not a plant is infested DEPENDS on whether or not the pesticide was applied; specifically, when pesticide is applied infestation is much lower than when pesticide is not applied.

Table 5- 2: Cases of independence and dependence between whether or not the pesticide is used and whether or not plants were infested

|Frequencies |Proportions (Percent) |

|Case 1 |Pesticide | | | |Pesticide |

| | |Yes |No | | | |Yes |No |

|Infested |Yes |12 |12 | |Infested |Yes |0.30 ( 30%) |0.30 ( 30%) |

| |No |28 |28 | | |No |0.70 ( 70%) |0.70 ( 70%) |

| | |40 |40 | | | |1.00 (100%) |1.00 (100%) |

| | |

| |Continued on next page |

| | |

|Table 5-1 continued | |

|Frequencies |Proportions (Percent) |

|Case 2 | |Pesticide | | | |Pesticide |

| | |Yes |No | | | |Yes |No |

|Infested |Yes |10 |34 | |Infested |Yes |0.25 ( 25%) |0.85 ( 85%) |

| |No |30 |6 | | |No |0.75 ( 75%) |0.15 ( 15%) |

| | |40 |40 | | | |1.00 (100%) |1.00 (100%) |

2 Hypotheses:

The Null Hypothesis or Ho for the test of independence is: Variable 1 is independent of variable 2.

The Alternative Hypothesis or Ha is: Variable 1 is not independent of Variable 2 or Variable 1 depends on variable 2.

For the preceding example, the Null Hypothesis would be: The application of pesticide is independent of whether or not the plants are infested.

Statistical Test:

We will use Systat™ to compute the tests. One of the advantages in using a computer statistical package is that, it not only computes the appropriate statistic, but it also provides a p-value (probability). To determine if you are to accept or reject Ho, you compare the p-value to alpha:

If the p-value ≥ alpha then you would ACCEPT Ho (variables independent)

If the p-value < alpha then you would REJECT Ho (variables dependent)

3 2x2 TEST of INDEPENDENCE – General Procedure

1) Determine what you are going to test.

2) Design the experiment.

a. What are the two variables?

b. What are the levels for the two variables?

c. What analysis should you use?

d. What are Ho and Ha? What would it mean if you accept Ho?

e. What would it mean if you reject Ho?

f. How would you conduct the experiment?

g. What statistical error should you avoid?

3) Collect data.

4) Compute test statistic.

5) Decisions: Compare p-value to alpha.

a. If the p-value ≥ alpha, ACCEPT HO.

b. If the p-value < alpha, REJECT HO.

i. Plot either column or row percents

6) Draw conclusion.

4 EXAMPLE 1: 2x2 TEST of INDEPENDENCE

A species of snake has two morphological variations, banded and unbanded. You think that the banding pattern may be an environmental response to the presence of brush (e.g., protective coloration). If you find that to be true, you will continue your research. You randomly sampled sites until you found 180 sites with snakes. For each site, you recorded whether or not brush was present and whether the snake was banded or unbanded.

1) Determine what you are going to test.

We want to determine if the snake pattern is related to the presence or absence of brush

2) Design the experiment.

a. What are the two variables? Snake and Brush

b. What are the levels for the two variables? Snake: Banded or Unbanded

Brush: Present or Absent

c. What analysis should you use? You are going to use a 2X2 test of independence to determine if the proportion of banded individuals varies between the locations (depends on the location). The test we will use is called the G-test or Log-Likelihood ratio test (Sokal and Rohlf, 1995); it is much better than Chi-square.

d. What are Ho and Ha? Ho is that whether a snake is banded or unbanded is independent of the presence or absence of brush. Ha is that whether a snake is banded or unbanded depends on the presence or absence of brush.

e. What would it mean if you accept Ho? It would mean that whether or not a snake is banded is not related to the presence or absence of brush.

f. What would it mean if you reject Ho? It would mean that there is a relationship between whether or not the snake is banded and the presence or absence of brush. There are two possibilities, the percentage of banded snakes could be greater in areas with brush or the percentage of banded snakes could be less in areas with brush.

g. How would you conduct the experiment? You will randomly sample sites until you find 180 sites with snakes. For each site, you will record whether or not brush was present and whether the snake was banded or unbanded.

h. What statistical error should you avoid? (see Table 5-3)

Table 5- 3: Determining which statistical error to avoid.

| |Conclusion |Action |What if I’m wrong? |Type of error |

|Accept Ho: Snake banded or unbanded |The presence or absence of |Abandon current |Missed an |II |

|is independent of presence or |brush is not related to the |research. |opportunity. | |

|absence of brush. |banding pattern of the snakes | | | |

|Reject Ho: Snake banded or unbanded |The presence of brush does have|Continue research |Do work for nothing |I |

|is dependent of presence or absence |something to do with the |efforts to find out|and falsely support | |

|of brush. |banding pattern of the snakes. |why. |your own idea | |

Conclude that the worse error is Type I so alpha will equal 0.025.

3) Collect data

Table 5- 4: Frequency of banded/unbanded snakes and presence/absence of brush for 180 sites with snakes.

|Data |BRUSH |

|SNAKE |Absent |Present |

|Banded |32 |46 |

|Unbanded |43 |59 |

4) Compute the test statistic

Use Systat™ 10.0 to analyze the data in Table 5-4. See page 5-7 for an example in how to use Systat.

5) Decisions: Compare p-value to alpha

Compare the value in the PROB column (p-value) to the alpha level you selected for this experiment.

If the p-value ≥ alpha, ACCEPT HO. If the p-value < alpha, REJECT HO.

For this particular example, we specified alpha=0.025 (see page 5-3). Since the p-value (0.879) is greater than alpha, we accept Ho where Ho is that: whether as snake is banded or unbanded is independent of the presence or absence of brush.

The statement you would make in the Results section of your paper would be as follows:

The 2x2 test of independence (G=0.023, df=1) showed that the banding pattern (banded or unbanded) was independent (p=0.879) of the presence or absence of brush.

6) Draw conclusion

The banding pattern doesn’t appear to have anything to do with the presence of brush in the environment.

5 EXAMPLE 2: 2x2 TEST of INDEPENDENCE – Where Ho is Rejected

We will use the same problem but with different data to illustrate what you should do if you reject Ho.

1) Collect data

Add 40 to the number of non-banded snakes found in the locations where brush was absent and reanalyze the data (see Table 5-5).

Table 5- 5: Frequency of banded/unbanded snakes and presence/absence of brush for 180 sites with snakes.

|Data |BRUSH |

|SNAKE |Absent |Present |

|Banded |32 |46 |

|Unbanded |83 |59 |

2) Compute the test statistic:

Use Systat™ 10.0 to analyze the data in Table 5-5. See page 5-7 for an example in how to use Systat.

3) Decisions: Compare p-value to alpha

Compare the value in the PROB column (p-value) to the alpha level you selected for this experiment.

If the p-value ≥ alpha, ACCEPT HO. If the p-value < alpha, REJECT HO.

For this particular example, we specified alpha=0.025 (see page 5-3). Since the p-value (0.013) is less than alpha, we reject Ho where Ho is that: whether as snake is banded or unbanded is independent of the presence or absence of brush.

The statement you would make in the Results section of your paper would be as follows:

The 2x2 test of independence (G=6.148, df=1) showed that the banding pattern (banded or unbanded) was dependent (p=0.013) on the presence or absence of brush.

Since we rejected Ho, we need to create a plot (Figure 5-4) of the column percents shown in Figure 5-3.

4) Draw conclusion

The percentage of banded snakes at a site is lower when there is no brush in the immediate surroundings. Perhaps the banded snakes experience higher mortality at sites without brush because they are more visible to predators.

6 Systat 10.0™ : Compute a 2x2 test of independence using the Log-likelihood ratio test (G test).

1) Create a data file: From the FILE pull-down menu, select NEW and then select DATA.

a. Create the variable SNAKE$. Double-click on the column name, VAR00001. In the next window, change VAR00001 to SNAKE$ (** Make sure you add the $ sign! **) then click on STRING for variable type. Then click on OK.

b. In the next column, create the variable, BRUSH$, which will also be a string variable.

c. Create a column to enter the frequencies. Change VAR00003 to FREQ and make the variable type NUMERIC.

2) Enter the data

a. In the SNAKE$ column, enter B for banded or U for unbanded.

b. In the BRUSH$ column, enter P for present of A for absent.

c. Enter the frequency in the FREQ column.

d. Your data should look like that in Figure 5-5.

e. **IMPORTANT** You must let the computer know that you are entering a frequency table instead of raw data/ Select FREQUENCY from the DATA pull-down menu. Then click on FREQ and click on ADD. FREQ should show up in the box labeled VARIABLE. Click on OK.

3) Data Analysis

a. From the STATISTICS pull-down menu, select TABLES, then CROSSTABS, and then TWO-WAY.

b. In the next screen (Figure 5-6),

i. Choose SNAKE$ as the row variable. Click on SNAKE$ in the box on the left and then click on ADD for the ROW VARIABLE.

ii. Choose BRUSH$ as the column variable.

iii. Select the types of tables to produce. In the TABLES section, select FREQUENCIES, ROW PERCENTS and COLUMN PERCENTS.

iv. Click on the STATISTICS BUTTON.

c. In the next screen (Figure 5-7), specify a G-test (Log-likelihood) by unselecting PEARSONS CHI SQUARE and selecting the LIKELIHOOD RATIO CHI-SQUARE in the RxC TABLES section. Then click on CONTINUE.

d. You will then be back to the screen shown in Figure 5. Click on OK to run the analysis.

4) Output from Systat™:

a. Systat puts the output in another window and then minimizes the window, To view the results, click on the bar (minimized window) at the bottom of your screen labeled “Untitled Systat Out..”.

b. The first part of the results tells you what file was used, when it was created and what variables were in that file (Figure 5-8).

c. The next part of the results is a frequency table (Figure 5-9). Compare your original frequencies to this table; the table values should match. If, not then you have made an error in your analysis.

d. The next two parts of the analysis are the row and column percentage tables If you reject Ho, then you will use these tables to chart and interpret your results.

e. The most important part of the output is at the end; it is a table showing the results of the statistical test (Figure 5-10). The G statistic is specified in the Value column (0.023) and the p-value is specified in the Prob column (0.879).

7 RXC INDEPENDENCE TEST

The RxC test of independence has two variables, one of which has more than two levels. The test is conducted in exactly the same way that the 2x2 test but, if you reject HO, you need to make additional tests. With a 2x2 test you are essentially comparing two proportions and the answer is straightforward. With an RxC test, you are comparing multiple proportions and it is difficult to determine which proportions differ from which.

There are two generally used procedures for comparing multiple proportions. Both of these procedures involve creating a series of 2x2 tables. So, the question may be, “Why not just do the 2x2 tables from the start instead of doing the overall test of independence?” The reason is that the type I statistical error rate goes up with the number of comparisons. In the first procedure, you figure out how many comparisons you would like to make and use a formula (Dunn-Šidák) to adjust your alpha level. The second is to conduct all possible pairwise comparisons between levels. The first procedure is often more useful because it allows you to combine levels to address a specific question so we will illustrate the technique described in Sokal and Rohlf (1995).

8 RxC TEST of INDEPENDENCE – General Procedure

1) Determine what you are going to test.

2) Design the experiment.

a. What are the variables?

b. What are the levels for the variables

c. What analysis should you use?

d. What planned comparisons do I want to make among the levels?

e. What are Ho and Ha?

f. What would it mean if you accept Ho?

g. What would it mean if you reject Ho?

h. How would you conduct the experiment?

i. What statistical error should you avoid?

3) Collect data

4) Compute the test statistic

5) Decisions: Compare p-value to alpha

a. If the p-value ≥ alpha, ACCEPT HO.

b. If the p-value < alpha, REJECT HO.

6) Conduct planned comparisons

7) Conduct any unplanned comparisons

8) Draw conclusion

9 EXAMPLE 3: RxC TEST of INDEPENDENCE

You have noticed that there are two varieties (morphs) of carapace coloration among individuals of a beetle species: bright red and dull red. You want to know if there are seasonal changes in the relative proportions of the two morphs. Specifically, you want to know if the proportion of bright to dull changes from Spring to Summer and if changes in morphs occur between early and late in a season. If, you find differences, you will continue your research and try to find out why there is a shift. Otherwise you will find another project.

1) Determine what you are going to test.

We want to determine if the proportion of Bright to Dull color morphs is related to the season.

2) Design the experiment.

a. What are the two variables? Color morph and Season

b. What are the levels for the two variables? Color: Bright or Dull

Season: Early Spring, Late Spring, Early Summer or Late Summer

c. What analysis should you use? You are going to use a RxC test of independence to determine if the proportion of Bright to Dull color morphs varies between the four seasons (i.e. depends on season). The test we will use is called the G-test or Log-Likelihood ratio test (Sokal and Rohlf, 1995).

d. What planned comparisons do I want to make among the levels?

Planned comparisons are between two group and you can make as many as a-1 comparisons where a=number of levels. In this case we could make 4-1=3 comparisons. The comparisons should reflect the questions you wish to answer. If you find that you cannot develop comparisons that address your primary questions, it usually indicates that you need to reconsider your experimental design.

The comparisons also have to be independent of each other. The simplest way to insure independent comparisons is to create the initial two groups using all of the variables and then splitting up the groups. For example:

Because we want to know if the proportion of color morph changes among spring and summer, the first comparison should be Spring versus Summer. We will get Spring by combining Early Spring and Late Spring. Likewise, Summer would be created by combining Early Summer and Late Summer. So the first comparison would be:

Early Spring+Late Spring versus Early Summer+Late Summer.

The next two comparisons would be made by breaking up each of the two groups into a comparison. The second and third comparisons would be:

Late Spring versus Early

Early Summer versus Late Summer.

You will notice that it is not possible to subdivide the groups anymore and that, by doing it this way, we always get the appropriate number of independent (e.g. a-1=3) comparisons.

Also, the initial two groups do not have to have equal number of levels. For example, let’s assume that we have a control and 3 dosages (Low, Medium and High) of a drug. The first comparison would be Control versus Drug (Low+Medium+High). The next comparisons might be Low versus Medium+High and then Medium versus High.

e. What are Ho and Ha? Ho is that whether a beetle is bright or dull colored is independent of the season. Ha is that whether a beetle is bright or dull colored depends on season.

f. What would it mean if you accept Ho? It would mean that the proportion of Bright to Dull beetles is not affected by season.

g. What would it mean if you reject Ho? It would mean that the proportion of Bright to Dull color morphs would vary with season.

h. How would you conduct the experiment? In each of four seasons you collected as many beetles as possible in the study site. For each beetle found, you will record whether was a Bright red or Dull red morph.

i. What statistical error should you avoid? (see Table 5-5)

Table 5- 11: Determining which statistical error to avoid.

| |Conclusion |Action |What if I’m wrong? |Type of error |

|Accept Ho: Bright versus Dull color |The proportion of Bright to |Find another |Miss out on a |II |

|morph is independent of season. |Dull color morphs has nothing |project |publishable finding.| |

| |to do with seasons. | | | |

|Reject Ho: Bright versus Dull color |The proportion of Bright to |Continue research |Wasted time and |I |

|morph is dependent on season |Dull color morphs does have |to try to determine|resources | |

| |something to do with the |why. | | |

| |season. | | | |

Because you have a limited budget, you conclude that Type I is the worst so alpha will equal 0.025.

3) Collect data

Table 5- 12: Frequency of bright red and dull red color morphs in early Spring, late Spring, early Summer and late Summer for 691 beetles.

|Data |COLOR MORPH |

|SEASON |Bright |Dull |

|E. Spring |29 |11 |

|L. Spring |273 |191 |

|E. Summer |8 |31 |

|L. Summer |64 |64 |

4) Compute the test statistic

Use Systat™ 10.0 to analyze the data. Run this in exactly the same way as you did for the 2x2 Test of Independence. See page 6-8 for an example in how to use Systat.

5) Decisions: Compare p-value to alpha

Compare the value in the PROB column (p-value) to the alpha level you selected for this experiment. If:

P-value ≥ alpha, ACCEPT HO.

P-value < alpha, REJECT HO.

For this particular example, we specified alpha=0.025 (see above). Since the p-value (0.000) is less than alpha, we reject Ho where Ho is that: Bright versus Dull color morph is independent of season.

6) Conduct planned comparisons if you have rejected Ho in step 5 .

Planned comparisons are really sets of 2x2 tables.

a. Spring versus Summer

i. The Ho is that frequencies of dull and bright morphs are independent of whether the insects were collected in spring or summer.

ii. To get the frequencies for spring, for each color morph, add the frequencies for early spring and late spring. Do the same for summer. (Table 5-13).

Table 5- 13: Combined Frequencies for 1st planned comparison

| |Bright |Dull |

|Spring (E. Spring + L. Spring) |29+273=302 |11+191=202 |

|Summer (E. Summer + L. Summer) |8+64= 72 |311+64=375 |

iii. Then run the test as you would a normal 2x2 test of independence. In this case, p < alpha so we reject Ho and conclude that the frequencies of dull and bright color morphs depends on whether the insects were collected in spring or summer (Figure 5-14).

Figure 5- 14: Hypothesis test for the first a priori comparison of Season (Spring and Summer) versus Color Morph (Bright and Dull)

iv. Because you rejected Ho for this comparison, you should compute column and row percentages (Figure 5-15) and graph the results. In this case, row percentages show what the proportions of bright and dull morph are with a season and column morphs show what the proportions of insects are present among the seasons for a given color morph. For our example, presenting the row percentages make more sense relative to our initial question (Figure 5-16).

b. Early Spring versus Late Spring

i. The Ho is that frequencies of dull and bright morphs are independent of whether the insects were collected in early spring or late spring.

ii. Again run the test as you would a normal 2x2 test of independence but do not include the frequencies for summer (Figure 5-17). In this case, p > alpha so we accept Ho and conclude that the frequencies of dull and bright color morphs do not depend on whether the insects were collected in early spring or late spring.

[pic]

Figure 5-17: Hypothesis test for the second a priori comparison, Early Spring vs Late Spring.

c. Early Summer versus Late Summer

i. The Ho is that frequencies of dull and bright morphs are independent of whether the insects were collected in early summer or late summer.

ii. Again run the test as you would a normal 2x2 test of independence but do not include the frequencies for spring (Figure 5-18). In this case, p < alpha so we reject Ho and conclude that the frequencies of dull and bright color morphs depend on whether the insects were collected in early summer or late summer.

[pic]

Figure 5-18: Hypothesis test for the third a priori comparison, Early Summer vs Late Summer.

iii. Because you rejected Ho for this comparison, you should compute column and row percentages and graph the results of the table that reflects percentage of bright and dull insects per season. In this case, the graph of percentage of color morphs shows that the number of dull color morphs is more abundant in early summer (Figure 5-19).

7) Conduct any unplanned comparisons.

Unplanned comparisons are any comparisons you wish to make AFTER you have completed the planned comparisons. These comparisons are usually conducted because you notice something interesting about your data that was not part of the original set of questions. Because these are comparisons that were not part of the original questions, you need to protect yourself against Type I error. To do this, you need to use a Dunn-Sidak correction (Sokal and Rohlf 1995) to determine a corrected alpha.

Dunn-Sidak corrected alpha=[pic] where k = the total number of comparisons (including the planned).

In this case, we noticed that late spring might be different than early summer and we would like to determine if the proportion of bright and dull morphs changes between those two portions of the seasons. Since we have already made three planned comparisons and will be making one unplanned comparison, k=4 and the corrected alpha will be:

[pic]

a. Late Spring vs Early Summer

i. For this comparison, do not include the values for early spring and late summer (Table 5-14)

Table 5- 14: Data for the unplanned comparison (deleted early spring and late summer)

|Data |COLOR MORPH |

|SEASON |Bright |Dull |

|L. Spring |273 |191 |

|E. Summer |8 |31 |

ii. Again run the test as you would a normal 2x2 test of independence but compare p to αNew. In this case, p < αnew so we reject Ho and conclude that the frequencies of dull and bright color morphs depend on whether the insects were collected in late spring or early summer (Figure 5-20).

[pic]

Figure 5-20: Test of hypothesis for the unplanned comparison, Late Spring vs Early Spring. The Dunn-Sidak corrected alpha=0.006.

iii. Because you rejected Ho for this comparison, you should compute column and row percentages and graph the results of the table that reflects percentage of bright and dull insects per season. In this case, the graph of percentage of color morphs shows that the number of dull color morphs is more abundant in early summer (Figure 5-21).

[pic]

Figure 5-21: The unplanned comparison, Late Spring vs Early Summer with respect to Color Morph (Bright and Dull)

8) Draw conclusion

During the summer months, there is a greater proportion of dull morphs than in the spring, The proportion of dull to bright color morphs does not differ between early and late spring but, in early summer, the proportion of dull morphs increased and then declines in late summer.

10 REFERENCE

Sokal, R. R. and F, J. Rohlf. 1995. Biometry: The Principles and Practice of Statistics in Biological Research, 3rd Ed. W. H. Freeman and Company, New York.

11 ON YOUR OWN

Problem: You are trying to find out if hair color (Black, Brown, Blond or Red) is independent of gender. If there is a dependency, you will try to determine why.

1) Determine what you are going to test.

2) Design the experiment.

a. What are the two variables?

b. What are the levels for the two variables?

c. What analysis should you use?

d. What are Ho and Ha?

e. What would it mean if you accept Ho?

f. What would it mean if you reject Ho?

g. How would you conduct the experiment?

h. What statistical error should you avoid?

Table 1: Determining which statistical error to avoid.

| |Conclusion |Action |What if I’m wrong? |Type of error |

|Accept Ho:. | | | | |

|Reject Ho: | | | | |

3) Collect data

Table 2: Frequency of male and female with different hair colors.

|Data |GENDER |

|HAIR COLOR |Female |Male |

|Black |55 |32 |

|Brown |65 |43 |

|Blond |94 |16 |

|Red |16 |9 |

4) Conduct the analysis and paste results below.

5) Decisions: Compare p-value to alpha

6) Conduct planned comparisons and make appropriate graphs if you have rejected Ho

7) Conduct any unplanned comparisons and make appropriate graphs.

8) Draw conclusion

-----------------------

Figure 5-10: Results of 2x2 Test of Independence

Figure 5-9: Systat output for 2x2 Test of Independence Part 2

Figure 5-8: Systat output for 2x2 Test of Independence Part 1

Figure 5-7: Statistics selection screen for two-way crosstabs

Figure 5-6: Screen for Two-way Crosstabs

Figure 5-5: Data entry in Systat

30

Figure 5- 16: First a priori comparison with Spring vs Summer with respect to Color Morph (Bright and Dull)

|Test statistic |Value |df |Prob |

|Pearson Chi-square |14.362 |1.000 |0.000 |

Figure 5- 4: Dependency between the presence or absence of brush and the percentage of banded relative to unbanded snakes at a habitat.

Figure 5- 12: Results of RxC Test of Independence: Systat "! 10.0

Early Summer

0

Season

Late Spring

Figure 5- 1: Results of 2x2 Test of Independence: Systat ™ 10.0

Figure 5- 2: Results of 2x2 Test of Independence: Systat ™ 10.0

Figure 5-3: Column percents from 2x2 Test of Independence

Figure 5- 15: Row and column percentages from Systat 10.0

Figure 5- 19: The third a priori comparison, Early Summer vs Late Summer with respect to Color Morph (Bright and Dull)

60

90

Percentage of color morph

DULL

BRIGHT

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