# calculating We can apply a statistical measure called goodness of fit to determine how well Mendel's theory

# calculatingWe can apply a statistical measure called goodness of fit to determine how well Mendel’s theory

actually explains his experimental results. This measure requires some arithmetic and careful thought, so bear with it. Here are the basic steps in performing the test.

- Determine the expected results: If we keep an open mind—in the face of Mendel’s actual results and the thousands of people who have written about them over the last 140 years or so—we have to admit that we can’t be sure the theory is right. But, just for the sake of argument suppose it is. Then our best estimate is that in Mendel’s experiment he’ll get 75% round and 25% wrinkled. In his actual experiment there were 7324 F2 plants, so we’d expect about 0.75(7324) = 5493 of these to be round and 0.25(7324) = 1831 to be wrinkled. We call these values the expected frequencies.
- Complete the goodness of fit table: The goodness of fit table summarizes all the basic calculations of the test. Table 7.1 is the goodness of fit table for this experiment. The first column lists all possible phenotypes. The second column, titled “Exp,” shows the expected frequencies in the experiment. The third column, titled “Obs,” gives observed frequencies actually obtained in the experiment. In the fourth column we calculate the difference between the previous two, Obs — Exp. In the fifth column we square the values in the previous column, (Obs — Exp)2. Finally, in the last column we divide the previous column by the expected frequencies.

EXAMPLE: Let’s recalculate the first row of table 7.1 to see how it’s done.

Fill in the first three columns. These values can be obtained in the preceding text.

Calculate the difference between observed and expected frequencies. In this case we have

5474 – 5493 = -19 (1)

Square this difference. That is, calculate -19^{2} = 361.

Divide this square by expected frequency. In this example we have

631

5493

= 0.0657 (2)

- After completing the goodness of fit table you are in position to calculate an important statistic called (the lower case Greek letter chi, pronounced “chi-square.”) The statistic is simply the sum of values in the final column of the goodness of fit table.

EXAMPLE: In Mendel’s experiment, = 0.0657 + 0.1972 = 0.2629.

Interpreting

The statistic measures the fit between theory and experiment. The larger gets, the less theory and experiment agree. If the value gets above a certain predetermined level, called the critical value, then we say that the experimental result is significantly different from the theory and therefore the theory is not supported. To develop a detailed understanding of why that’s true requires mathematics beyond the scope of this course, but everyone in any biological science or related field should have at least a rudimentary understanding of how statistics like this are used.

So, where does our value of = 0.2629 fall relative to the critical value? Without further study you cannot be expected to find the critical value for yourselves, so I’ll provide it. In Mendel’s experiment, the critical value is 3.841^{1} . Since 0.2629 is less than 3.841, we conclude that the experimental facts fit the theory. (In fact, in this case is almost too small, meaning the facts fit almost too well, leading some to speculate that Mendel actually “adjusted” his data a little.)

Testing Mendel’s first law in corn

It’s impossible to perform the kinds of experiments Mendel did with humans because we breed too slowly. So like Mendel we will use a plant—corn, or to be more precise Zea mays—to test his theory of genetics. In this analysis we will focus on a single easily identifiable trait, namely kernel color.

Some corn kernels are yellow/white and some are purple. The theory states that these traits are determined by a single gene in a Mendelian fashion. Yellow is the recessive trait, so yellow kernels have genotype aa and purple kernels may have genotype AA or Aa. You can’t see any difference between AA and Aa; they are equally purple.

- Here is a photo of a single ear of. This ear contains all F2 individuals, meaning it was obtained by crossing two corn plants thought to be heterozygous for kernel color. Each kernel is an individual. (It’s a seed capable of growing into an independent corn plant with a unique genotype.)
- Choose a single kernel at the fat end of the cob. It doesn’t matter which kernel you choose as long as it’s on the wider end.

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^{1}For those interested in where this number comes from, I’ll try to give a quick motivation. We started by assuming that Mendel’s theory was correct, just for the sake of argument. That assumption is called the null hypothesis. If the null hypothesis is true, then mathematicians can show that the probability one would get a value equal to or greater than the critical value is less than a predetermined level of significance. In this example I set this level of significance at 0.05, meaning that if we were to get an experimental result with a probability less than 5% under the null hypothesis, I’ll conclude that the null hypothesis is not likely to be true.

- Note that the kernels run more-or-less in rows down the cob. Starting with your chosen kernel, count the number of purple and yellow kernels in your first kernel’s row. If you get to a point where you can’t tell where the row goes just make your best guess and move on. It really doesn’t matter as long as you don’t count any kernel twice. Keep careful track of your tally in Excel or on a separate sheet of paper.
- When you have finished with that row, skip to the row above the kernel you started with and note the color of kernels in that row.
- Continue in this fashion until you have counted at least 100 kernels. You may do more than

100 if you wish.

Table 7.2 Goodness of fit table for the corn color analysis

Phenotypes | Exp | Obs | Obs-Exp | (Obs – Exp)^{2} | (Obs – Exp)^{2}/Exp |

Purple | |||||

Yellow |

- Use your data to complete the relevant goodness of fit table (table 7.2). Recall that we expect 75% of the kernels to be purple and 25% to be yellow. So, to find the expected frequency of purple multiply the total number of kernels you counted by 0.75. Calculate expected frequency of yellow by multiplying the total by 0.25.
- Calculate from table 7.2.
- In the space below, state the null and alternative hypotheses of this experiment.
- The critical value in this situation is 3.841, the same as in the example. Do your data support Mendel’s theory or not? Explain why or why not in detail in the space provided.

Single trait test cross

Mendel’s exact thought process leading to his discovery remains mysterious because all we really have is his 1859 paper, and that has been “cleaned up,” so to speak. However, given his presentation and it similarity to other discoveries, the following sequence of events is probably pretty close to reality:

- After years of growing peas in his garden for the other monks, Mendel notices certain repeatable patterns when he crossbreeds plants with different traits. These observations lead to more careful trials that coalesce into the observations he published in his paper—the loss of on parent’s trait in the F
_{1 }and its reappearance in a 3:1 ratio on the F_{2}for seven different traits. - Mendel then devises his so-called “Law of Segregation” to explain these observations. In short, he shows that all his observations can be deduced from three simple facts: (1) each trait is determined by two alleles, (2) one allele comes from each parent, and (3) one allele dominates the other in the heterozygote.
- Mendel is smart enough to know that so far his argument is relatively unconvincing. He saw a pattern and made up a story to “explain” it. Big whoop. He knows that his hypothesis should only be taken seriously if it explains data or observations that it was not constructedto explain. . In other words, it should correctly predict the outcome of experiments never before performed.

So, to test his hypothesis, Mendel’s needed to plan a new experiment, one for which his hypothesis would predict a measurable outcome. He did just that in what we now call a test cross, in which Mendel backcrossed F_{1 }plants with the wrinkled parents. The procedures below will walk you through Mendel’s test cross in corn.

- Obtain an ear of corn from the pile labeled “test cross.” This ear was obtained by backcrossing purple F
_{1}(Aa) plants with their yellow (aa) parents, as Mendel did with his pea plants. - Before counting any kernels, use Mendel’s hypothesis to predict the results of such a cross. Place your prediction and show your work in the space below.
- As before, count at least 100 kernels on the test cross cob, starting from the fat end. Tally the number of purple and yellow kernels in your sample.
- In the space below, construct and complete a table for this cross.

- Calculate and interpret the statistic for this experiment in the space below. The appropriate critical value is still 3.841.

Testing Mendel’s Law of Independent Assortment

Mendel didn’t stop experimenting after he discovered the Law of Segregation. This law governs the inheritance of single traits like perfect pitch, seed shape in peas and perhaps corn kernel color. But Mendel became interested in how pairs of traits were passed from parent to offspring. So, in one experiment he crossed a true-breeding variety of pea that had round seeds and a yellow seed leaf with another true-breeding variety with wrinkled seeds and a green seed leaf.

Today we would write the genotype of the round/yellow variety as AABB, where the A alleles determine seed shape and B determines seed color. Similarly, the wrinkled/green plant has genotype aabb. Based on theory, what would we expect the F_{1} genotypes to be? According to Mendel’s second law (Law of Independent Assortment) the genes for seed shape are passed down independently of genes for seed color. So, when the AABB parent makes gametes, it places one of the two A and one of the two B alleles into its gametes. Therefore, its gametes have genotype AB. The other parent similarly makes ab gametes. To make a baby plant a AB gamete fuses with a ab gamete, so the resulting offspring has genotype AaBb. Therefore, we expect nothing but round/yellow plants in the F_{1}, which is exactly what Mendel observed.

At this point Mendel again self-pollinated two F_{1} plants to produce an F_{2} generation. What do we expect from this AaBb x AaBb cross, called a dihybrid cross? Mendel postulated that to make gametes, a plant with genotype AaBb places one of its alleles for seed shape (A or a) and one of its alleles for seed color (B or b) into its gamete. So it can make four different types of gametes

— AB, Ab, aB and ab — with equal numbers of each type.

If Mendel’s theory is correct then we can build a Punnett square for this situation just like we did before, by listing all gamete types from one parent along the top and gametes from the other parent along the left side. In this case we have a 4 x 4 Punnett square. Then in each cell we combine alleles from the head of the columns and rows, by convention writing the A alleles before B and capitals before lowercase. The result is this:

AB Ab aB ab

AB | AABB | AABb | AaBB | AaBb |

Ab | AABb | AAbb | AaBb | Aabb |

aB | AaBB | AaBb | aaBB | aaBb |

ab | AaBb | Aabb | aaBb | aabb |

If you count up the phenotypes from this table you should see that 9/16 (= 0.5625) are round/yellow, 3/16 (= 0.1875) are round/green, 3/16 are wrinkled/yellow and 1/16 (= 0.0625) is wrinkled/green, which gives us our expected values from theory.

Table 7.4 Goodness of fit table for test of Mendel’s second law.

Phenotypes | Exp | Obs | Obs-Exp | (Obs – Exp)^{2} | (Obs – Exp)^{2}/Exp |

Purple/Round | |||||

Purple/Wrinkled | |||||

Yellow/Round | |||||

Yellow/Wrinkled |

When Mendel performed his experiment he actually observed 315 round/yellow, 108 round/green, 101 wrinkled/yellow and 32 wrinkled/green F_{2} plants. Does this fit the theory? Mendel ended up with a total of 556 F_{2} plants. Therefore, 56.65% (315/556 = 0.5665) were round/yellow. Similarly, 19.42% were round/green, 18.17% were wrinkled/yellow and 5.76% were wrinkled/green. Are these numbers close enough to support Mendel’s second law or not?

Again, we don’t just blurt out “yes” or “no” because we feel it’s right; Nature doesn’t care how we feel. Instead, we use a rational procedure, the goodness of fit test, to answer this question. The goodness of fit table for this analysis is given in table 7.3. To get we sum the values of the last column, so = 0.4695. For this analysis the critical value has risen to 7.815, so again our value is well below the critical value, and Mendel’s theory is supported.

In the following procedures you will test to see if two traits in corn obey Mendel’s Law of Independent Assortment. One trait — kernel color — you’ve already analyzed. The other trait is something you may have noticed — kernel shape. Some kernels are wrinkled and some are round. This is another trait that appears to follow Mendel’s first law. Wrinkled kernels have genotype bb and round ones could be either BB or Bb. These corn cobs were produced by double heterozygous corn plants AaBb, with genotype purple/round.

- Return to your F
_{2}cob. As before, choose a starting kernel on the fat end of the cob. Work your way down the cob tallying the number of each phenotype in the row. Possible phenotypes are purple/round, purple/wrinkled, yellow/round and yellow/wrinkled. Tally at least 100 kernels total. - When you’ve finished collecting data, place your results into table 7.4 and then complete that goodness of fit table.
- Calculate from table 7.4.

=

- The critical value is 7.815. Do your data support Mendel’s theory or not? Explain in detail in the space provided.