3. Extensions to Mendelian Inheritance
Chi Square Analysis
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Okay. So this practice problem, we're gonna be walking through the steps to using a Chi Square test. So here's the example question. So you have a purple plant and you think it's heterozygous. You don't know. Right? You think. So if it's not heterozygous, what could it also be? Could also be homozygous. Right? A a or a a, uppercase, lowercase, or just all uppercase. But you think it's heterozygous, and you wanna know for sure. So, what you wanna do is you wanna breed it with a homozygous recessive. Now, this has a fancy name. It's called a test cross. So you may see that in a question that's similar to this. It says, do a test cross. Test cross is you make whatever you have with a homozygous recessive. So that's a test cross. So this is what you're doing. So you have this purple plant, you think it's heterozygous, so you're going to take a, a white plant that you know is homozygous recessive and mate them together. And it says, okay, after you do that mating, you produce a 120 offspring, 55 are purple and 65 are white. Was your plant heterozygous? So, how do we actually go about doing this and using a Chi square? Well, the first thing that you have to do is so you know your these are the observed numbers. Right? Because you, observed this. You did an experiment. You have a 100 20 offspring, 55 are purple, 65 are white. So these are your o's. Remember the the formula for, Chi square, I'll show it down here again, but this is this is it. O minus e squared minus e, or should the second anyway. So you have these numbers, gave them to you in the problem. But now the very first thing you have to do is determine what the expected numbers are gonna be. So if you cross a heterozygous with a homozygous recessive, what do you get? First thing you have to do is just a Punnett square. So here's your heterozygous, here's your homozygous. Each allele gets its own column or row. Now, when you mate these together, right, you get one half are heterozygous, and these would be purple, And you get 1 half homozygous, and these would be white. So if you have a 120 offspring, remember, which is what the question gave us. How many are gonna be purple and how many will be white, expected? Back out. Remember we're working with expected. Well the Punnett square tells us that half will be purple and half will be white. So half of 120, 120 divided by 2 is 60. So here are the expected numbers. We got these from the Punnett square that we did if we use the heterozygous and homozygous. Half are going to be purple, half are going to be white. So out of 120, 60 are purple, 60 are white. These are your expected numbers and that's the very first step to doing a Chi square. So this is great. So we have our observed, 5560 5. We have our expected, 6060. So now, we can use the the chi square formula. Do you remember? X squared equals o minus e to the second over e. So you say, you may say, okay, well, I have this formula, but do I use 55? Do I use 65? How do I do this? Well, you actually do it for each class. So, unfortunately, I wrote red here for some reason, but what I'm trying to say is purple, and you do it for every number. So purple is 55 with the observed and expected are here. So we got we actually observed 55 purple. We expected from the Punnett square to get 60, then if we put it into here, o minus e to the phrase is 25, and then if we divide it by e, that's 0.42. Then because this formula has this thing in it, the sum, then we do the same thing for the white. 65 is observed, 60 was expected, 25.42. And then we add these together to get our chi square value of 0.84. So does that make sense? You get everything in that step. Right? We we, the problem gave us our observed, we calculated our expected through a Punnett square, and then we plug these numbers into a chart like this so that we can solve this formula here. Now, you're welcome if you're given this problem, you know, in a test or quiz to make your own graph. I really highly suggest that you do, so you don't get confused. And so you just solve the problem and now we have our chi squared value. So now what do you do with that chi square value? What does point 84 actually mean? Well, when you have this number, there's a special table called a chi squared distribution table. Now if you're given a question like this in a test or quiz, you will be given this table. You don't have to memorize it. You'll be given it. And so, you use the Chi squared to determine whether or not your hypothesis is true. Remember, our hypothesis or what we thought was that the plant was heterozygous. So how do you use this table? So this is what the table looks like and it looks a little overwhelming. So how do you actually use the table? Well, the first thing you do is you calculate what is called the degrees of freedom and that's here. That determines what row you're going to use or find the point a for on. So how do you calculate that? Well, that formula is the degrees of freedom or d f equals the number of variables minus 1. In this question, there are 2 variables. How are there 2 variables? Because we have purple and white looking at 2 colors. So in this question, our degrees of freedom is 2 minus 1, the number of variables minus 1, and that equals 1. So now we have our degrees of freedom and we know that we are going to be using this row. If our degree of freedom was 6, we'd use this 1. If our degree of freedom was 9, we'd use this 1, but it's not, it's 1, so we are using this row. This row right here, this is the row we're looking at. So now you have your degrees of freedom, you know what row you're using. Now you use your chi squared value, which if you remember, ours was 0.084. So what you do is you look down this row and you figure out where would 0.84 be. So 0.84 is not between this. It's not between 0.02 and 0. 6. It's not between 0.64 and point and 10.83. Where it is is 0.86 falls somewhere between 0.046.07. So this is where this it fits in. So these are the 2 rows you're looking at right here. And you say, okay. Well, now I have so many numbers. Now what is this 0.46 and 1.07 stand for? You don't actually need to know these numbers. These numbers are really not important for anything right now. What you need to know is finally what the p value is. Okay? So these two values sit above these 2p values, 0.5.3. Scroll down a little. See? And you say, okay, now here's another number. What do I do with this? Well, what 0.5.3 means is 50 30%. So what are you so what does that mean? Right? So this means that for this problem, there is a it's 50 to 30%. So I don't want to mess you up. So I was not going to use that explanation I thought I was going to use. Instead, we're going to move to the 4th the 4th step. So I get you're confused, but what you did here, just a quick review on step 3, you started with your Chi square, you calculated your degrees of freedom, this was 1, you looked on row 1 to where 0.84 were fit. You found these two numbers and you looked down to see what numbers they fit between on the p value here, and you found this is 0.5 and point 3, which equals 50 to 30%. Now, what do you do with 50 to 30%? What does that actually mean? And this is when you determine whether you accept or reject your null hypothesis. So the null hypothesis is not, I think this is heterozygous. Instead, the null hypothesis states that there is no difference between measured and predicted. So this would be observed and expected. So that means that the 55 number that we got is really no different than 60. I mean, it is different but it's not statistically, it's not mathematically know, it might you know, it might be a little bit off, but it's close enough that you can say it's the same thing. So this is your null hypothesis. Now every null hypothesis and every experiment you're gonna do is gonna be exactly the same. It's gonna say that the observed and the expected are not different from each other. And so, in this problem, our null hypothesis is that the 55, guess I changed colors here, the 55 purple and 65 white plants is close enough to the 60 purple and 60 white plants, and therefore, they are not different. So step 4 is to determine whether or not that statement is true or not. Is it true that they're not different or is it false and they are different? So generally, for this, you accept the null hypothesis. Now, I'm saying except if you are a math nerd, or if you are a statistician, or you have a really strict statistician geneticist professor, they're not going to want you to use the word accept. I use it here, because I think it's easier to understand. What they're going to want you to use is fail to reject. Remember, these are the exact same things. I can't determine which one your professor wants you to use, but I suggest that you actually ask them this. Do they prefer the word accept, or do they prefer the word fail to reject? I'm going to use accept, because I think it's much less confusing than fail to reject. But just know that in other statistics classes, and then, you know, certain professors classes, they're not gonna want you to use that term. But it for me, I'm using them interchangeably. So is it true that our observed and expected values are pretty much the same? So I accept that statement. So I said, yeah, they're close enough. It, you know, it's the observed and the respected are close enough. I accept that as true. If my probability, remember the p value from above is greater than 5%. Now, ours we have 30 to 50%. So that's much greater than 5. So in this problem we are accepting this, this null hypothesis that they are not different. You would reject it if it's less than 5%. And you would say, okay, no. These are very different numbers. 5560 are way too far apart. This this null hypothesis is false. I'm rejecting it. So, like I said, but for this question, it is 30 to 50%. Therefore, we are accepting the null hypothesis. And so the 5th thing is to remember what that actually means. So a lot of times people get through, you know, they power through the math, they know whether or not they're accepting or rejecting the null hypothesis, and they're like, great, I'm accepting this null hypothesis and then they forget what the question was answering. So or asking. So this here, super important step and it actually a lot of people will miss a question on a test or a quiz, because they skip this step. They just say, okay, yeah, I'm accepting the null hypothesis. But you have to go that one step further and remember what you were doing in the first place. So remember that we are saying that the null hypothesis means that the observed and the expected aren't different. They are the same, essentially. And so in this question, we were talking about whether or not the purple plant was heterozygous or homozygous, and we predicted that it was heterozygous. We thought it was this, genotype. Right? And so if we accept the null hypothesis, that means that we are 95% confident, and then how I got this number, I'll explain in one second, but we're 95% confident that the purple plant was indeed heterozygous, because our observed value from our mating, which was 120 and it was 5565, was not different than the expected value of 60 and 60. If it was different, then we would say, no. This isn't heterozygous. But because these two values are considered not different statistically, then we are confident that this plant is heterozygous. So this is the super important step. Do not forget this step. Remember, if you accept or reject the null hypothesis, go back to the question, the very original question and remember what you were testing to begin with And write your answer that way, not just I accept or reject the null, because you'll likely be kind of wrong for that. Now, for those who are interested or who have professors that are really math heavy, notice here that I said 95% confident. I'm not a 100% confident. Of course, in math, you're never a 100%, but I am 95% confident. So why did I put 95%? If you're interested, keep listening. If you're not, you can totally tune out. That's fair. The reason I did 95 is because of this here, specifically, these numbers. So I said that we accepted the null if it was greater than 5%, and I rejected the null if it was less than 5%. Now, 95 plus 5 equals 100. So this is where we're this is where we're not getting a 100%. We're saying that there's a 5% 5% chance I could be wrong because I left that up for error. Right? I said, I accepted this if the probability was greater than 5%. Now, I could have accepted it if it was greater than 2%. And in that case, it would have been 98% confident. So I'm taking I'm giving myself a little wiggle room here. I'm saying because I put 5%, which is typically what people do, I'm saying that, okay. Well, I could be wrong 5% of the time, but 95% is good enough, And so that's what I'm going to use. Now, I could do 0.0001% here and be, like, really super confident. But if I do that, I may actually miss something. And I may, you know, say, oh, these are totally different things when they're not. Just because I didn't give that myself that error room that you need when dealing with, you know, natural occurring things. Because there could just be an extra plant, and it doesn't mean anything. But if I really was too restrictive, I would say, oh, that, you know, that plant is one too many, and that's it. They're not the same. I I reject this null when really you should have accepted it and you were wrong. So 95%, people like to give themselves 5% error room, like to give their plants or their experiments 5% error there. And so this is why I said I was 95%, not a 100. But for statistics, 95 is close enough. It's actually very good. When you're 95% confident, you're fairly well confident. And so, so that is how you do a Chi Square. So realize it's a lot of steps. It's 5 steps. I would say the 5th is the most important. Don't do all the math and forget this last step. And, yeah, so I get it's a, it's a long question but you're going to see questions like this on a test. Like, I guarantee you, you'll see a Chi Square question. So, make sure you really understand this, before moving on to other things. So, with that, let's now move on.
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