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A quiz asks me to calculate the likelihood ratio for 50 heads out of 100 coin tosses. I understand that a coin can be either fair or not fair and that a fair coin has a probability of 0.5 for heads.
The quiz does not give me the probability of the unfair coin so how can I be expected to calculate the likelihood ratio? Is there a convention I should be using?

[Update] The quiz also asked me for the case of 5 heads out of 10 tosses. In this case I used p=0.4 as the alternative hypothesis and correctly calculated 1.23 as a ratio. When I try this technique for 50 out of 100 I get 1.02 but this is marked as incorrect.

My efforts with the code in the test are as follows

n<-100 #set total trials
x<-50  #set successes
H0 <- 0.5 #specify one hypothesis you want to compare with the likihood ratio
H1 <- 0.49 #specify another hypothesis you want to compare with the likihood ratio (you can use 1/20, or 0.05)
R1<-dbinom(x,n,H0)/dbinom(x,n,H1) #Returns the likelihood ratio of H0 over H1

Which outputs 1.02 for R1 at 2 decimal places.

I am trying to understand why this is wrong. Could the quiz be wrong? The comment in the code that I can use 0.05 as H1 is especially confusing.

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  • $\begingroup$ Presumably the intent is not that you choose a specific value but that you use an algebraic symbol. $\endgroup$
    – Glen_b
    Dec 29, 2021 at 10:51
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    $\begingroup$ In your code, H1 is $0.49$ rather than the $0.4$ you state: perhaps that's why your answer was marked incorrect. As remarked in the answers, it's more informative to use a spectrum of possible alternatives, such as setting H1 <- seq(0, 0.50, by=0.01). This will produce 51 ratios that you can easily plot via plot(H1, R1, log="y"). $\endgroup$
    – whuber
    Dec 30, 2021 at 20:44
  • $\begingroup$ When the quiz has n = 10 I use H1= 0.4 When the quiz has n = 100 I use H1=0.49 The answer is scored wrong in the case that n = 100. Why? $\endgroup$
    – Kirsten
    Dec 30, 2021 at 20:47
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    $\begingroup$ We can't tell you anything about the quiz scoring: we don't have a copy of the quiz or its scoring rubric. This is a question to raise with your instructor. $\endgroup$
    – whuber
    Dec 30, 2021 at 20:51
  • $\begingroup$ "A quiz asks me to calculate the likelihood ratio for 50 heads out of 100 coin tosses" It would also help if you quote the quiz exactly. $\endgroup$ Dec 30, 2021 at 22:42

3 Answers 3

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You can compute the likelihood ratio as a function of $p$

$$\Lambda_{LR} = \frac{\mathcal{L}(p\vert \text{H} = 50, \text{T} = 50)}{ \mathcal{L}(0.5\vert \text{H} = 50, \text{T} = 50)} = 2^{100} p^{50} (1-p)^{50}$$

ratio as function of p


In the context of a likelihood ratio test where the alternative hypothesis is a composite hypothesis you choose the value of $p$ for which the likelihood of the alternative hypothesis is maximized. So then the likelihood ratio, in the case of 50 tails and 50 heads is equal to 1.

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  • $\begingroup$ I am so confused. Looking at the chart that would make p = 0.5 which is the same as that of the null hypothesis $\endgroup$
    – Kirsten
    Dec 29, 2021 at 21:09
  • $\begingroup$ @Kirsten all of the region from $0$ to $1$ except the point $p=0.5$ is the alternative hypothesis. $\endgroup$ Dec 29, 2021 at 22:43
  • $\begingroup$ I updated the question to better explain my confusion. Perhaps it is a faulty quiz? $\endgroup$
    – Kirsten
    Dec 30, 2021 at 20:36
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It depends what the null and alternative hypotheses are. I assume the null is that the coin is fair ($Pr($heads$)= 0.5$), which is a simple null.

If the alternative is simple too, just write it generically as $Pr($heads$)=p_1\neq 0.5$ and proceed with constructing the likelihood ratio working with $p_1$.

If the alternative is composite of the form $Pr($heads$)\neq 0.5$, then you can proceed by setting up the generalized likelihood ratio, suping over the parameter space $[0,1]$. In your case, under 50 heads of 100 tosses, the likelihood is maximized at $Pr($heads$)= 0.5$, so the ratio of suped likelihoods must be 1 (or the generalized likelihood ratio test statistic, which logs this ratio, would be 0).

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Your question is from the coursera course "Improving your statistical inferences" by Daniel Lakens.

In the instruction PDF on the previous lecture there is more information, which is missing here:

Earlier we mentioned that with increasing sample sizes, we had collected stronger relative
evidence. Let’s say we would want to compare L(θ = 0.4) with L(θ = 0.5).
Q4: What is the likelihood ratio for 5 out of 10 heads?
Q5: What is the likelihood ratio for 50 out of 100 heads?
Q6: What is the likelihood ratio for 500 out of 1000 heads? 

That means, you need to calculate the likelihood ratio between L(θ = 0.4)/L(θ = 0.5) for the case of observing 5/10, 50/100 or 500/1000 heads. The likelihood ratios are therefor 1.23, 7.7 and 731784961.36. It is indeed not well formulated in the quiz, I was also confused at first as to what I had to do.

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