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28

The logit is a link function / a transformation of a parameter. It is the logarithm of the odds. If we call the parameter $\pi$, it is defined as follows: $$ {\rm logit}(\pi) = \log\bigg(\frac{\pi}{1-\pi}\bigg) $$ The logistic function is the inverse of the logit. If we have a value, $x$, the logistic is: $$ {\rm logistic}(x) = \frac{e^x}{1+e^x} $$ Thus (...


25

The odds is not the same as the probability. The odds is the number of "successes" (deaths) per "failure" (continue to live), while the probability is the proportion of "successes". I find it instructive to compare how one would estimate these two: An estimate of the odds would be the ratio of the number of successes over the number of failures, while an ...


24

On another thread there is a much broader answer by @gung that also deals with related technical issues such as the odds ratio, but I am going to stick to the topic at hand: how to interpret odds, and particularly the formulation "$a$ to $b$". As a beginner's question, it's worth thinking how "odds" are expressed in everyday speech (especially in betting ...


22

The advantage is that the odds defined on $(0,\infty)$ map to log-odds on $(-\infty, \infty)$, while this is not the case of probabilities. As a result, you can use regression equations like $$\log \left(\frac{p_i}{1-p_i}\right) = \beta_0 + \sum_{j=1}^J \beta_j x_{ij}$$ for the log-odds without any problem (i.e. for any value of the regression coefficients ...


17

The odds is the expected number of "successes" per "failure", so it can take values less than one, one or more than one, but negative values won't make sense; you can have 3 successes per failure, but -3 successes per failure does not make sense. The logarithm of an odds can take any positive or negative value. Logistic regression is a linear model for the ...


14

The total number of assignments among $2n$ people, where nobody is assigned to themselves, is $$d(2n) = (2n)!(1/2 - 1/6 + \cdots + (-1)^k/k! + \cdots + 1/(2n)!).$$ (These are called derangements.) The value is very close to $(2n)! / e$. If they correspond to perfect pairings, then they are a product of disjoint transpositions. This implies their cycle ...


10

A key step is to make sure people understand why log-odds-ratios are useful. To help motivate log-odds-ratios, try the tale of two principals: High School A reduced the dropout rate from 10% to 5%, a dramatic 50% decrease! High School B increased the graduation rate from 90% to 95%, a modest 5.5% increase. The first principal was lauded by the NYTimes ...


9

I think that the first and biggest hurdle is making sure that people indeed understand logistic regression and what an odds ratio actually is. If they get that far, you simply need to explain that proportional odds models take logistic regression one step further to account for ordered categorical responses. A naive approach could be to run a logistic ...


9

I don't have the required reputation to vote, so I'll add it as an answer instead. I fully agree with what @whuber said. The typical approach in this kind of study is to a priori declare a level of significance. Quoted from the article, the authors indeed do this, To accommodate the many comparisons made, two-tailed P values of less than 0.01 for the ...


7

A probability lies between 0 and 1; I presume you're familiar enough with it that we don't need to define probability, but please clarify if you do require a basic definition. [So if I say something like "The probability of at least two heads in three tosses of a coin is 1/2" or "the probability that I miss the bus tomorrow is 1/10" I presume the sense of ...


7

You are right that the "$\ln(p/(1−p))$" is the log odds. That means all you have to do to calculate the log odds is plug in the values you want for your $X$'s and do the arithmetic. Here is the calculation for #1: \begin{align} \text{log odds}(X_1 = 3; A) &= 0.3211 + 0.27\times X_1 + 0.732\times X_2 \\ &= 0.3211 + 0.27\...


7

I was quite impressed by the elegance in @whuber answer. To be honest I had to do a lot of acquainting myself with new concepts to follow the steps in his solution. After spending a lot of time on it, I've decided to post what I got. So what follows is an exegetical note to his already accepted response. In this way there is no attempt at originality, and my ...


7

You are right. If the book said that, it is wrong. I do wonder if it is a typo or a poorly phrased passage that lends itself to misunderstanding, though. As you show, $\exp(\beta_0 + \beta_1X)$ is the odds of 'success' predicted by the model. The odds ratio associated with a $1$-unit change in $X$ is $\exp(\beta)$. You can also think of is as the factor ...


6

The odds you have are in decimal format, which the bookmaker calculates as: $$ d_E = \frac{1}{p_E + o_E} $$ where $d_E$ is the decimal odds for event $E$, $p_E$ is the bookmakers estimated probability of event $E$, and $o_E$ is the over-round which the bookmaker adds to the decimal odds for event $E$. The over-round effectively reduces the odds, making ...


6

Under a simple Bernoulli experiment model setting, we may view each flight travel as an "experiment" with "success" rate $p = 2.6/1,000,000$ and "failure" rate $q = 1 - p$. Let $X$ denote the random variable that the number of flights a particular person travelled to suffer a first accident. Clearly $X \sim \text{Geometric}(p)$, and the odds you are asking ...


6

I believe the mistake you are making is the connection between the coefficients and the probability. Defining the linear predictor (i.e. the estimated log odds) as $\eta = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ...$ You are right that $\exp(\beta_1)$ represents the fold difference in odds associated with a one unit increase in $x_1$. So for age, a one ...


5

I've been caught in these circular loops many times. I think the bottom line of your question is: "In summary, how can each flight be considered an independent variable, but also have an increasing chance of something occurring?" As you said, we can use the geometric distribution, interpreted as the number of trials ($k$) (or flights) before the first ...


5

Summary The question misinterprets the coefficients. The software output shows that the log odds of the response don't depend appreciably on $X$, because its coefficient is small and not significant ($p=0.138$). Therefore the proportion of positive results in the data, equal to $100 - 19.95\% \approx 80\%$, ought to have a log odds close to the ...


4

If the regression coefficient of your logistic regression is 1.76 on the logit-scale, then the odds ratio for 1 unit increase in temperature is $\mathrm{OR_{+1}}=\exp(\beta) = \exp(1.76)\approx 5.81$, as you already stated. The odds ratio for an increase in temperature for $a$ degrees is $\mathrm{OR_{+a}}=\exp(\beta\times a)$. In your case, $a$ is 2 and 3, ...


4

It sounds like you have a lot of data, but no model yet. What do you learn from the $1$ run scored from a ball? There are a lot of possibilities. You might learn just what happened on that ball. You might learn something about the batter. You might learn something about the bowler. You might learn something about the overall skill levels of the teams. ...


4

This can be reported as: The odds of death are reduced by 14% with every 10 year increase in age, i.e. the odds ratio was 0.86, after controlling for other factors. or: After controlling for other factors, the odds ratio for 10 year increase in age was 0.86, i.e. the odds of death were reduced by 14% with every 10 year increase in age. ...


4

If the goal is to obtain 7+ eyes and there is no preference about the total score otherwise - then there is no difference between rolling 3 vs rolling 2 and re-rolling the lower one when their sum is lower than 7. Think about it like this: You roll 3 dice. You look at the first 2 of them. If they do not sum to 7 you look at the 3rd. But you will always ...


4

Here are two pairs of probabilities with $\operatorname{logit} p_1 - \operatorname{logit} p_2$ equal but $p_1 - p_2$ not equal: $p_1 = \tfrac{3}{4}$, $p_2 = \tfrac{1}{2}$ $p_1 - p_2 = \tfrac{1}{4}$ $\operatorname{logit} p_1 - \operatorname{logit} p_2 = \ln 3$ $p_1 = \tfrac{9}{10}$, $p_2 = \tfrac{3}{4}$ $p_1 - p_2 = \tfrac{3}{20}$ $\operatorname{logit} ...


4

It's easy to get confused between the two different types of probabilities that you face in this type of study. One is the probability, before you've run any tests, that someone with such a tumor has the malignant type: the 70% prevalence of the malignant form for this type of tumor. It is prior, apriori probability. The second is the probability, after you'...


3

The point of the odds ratio interpretation in logistic regression is that logistic regression is a linear model for the log odds of success. So a unit increase in an explanatory variable will result in increase or decrease of the predicted odds by a factor of $\exp(b)$, regardless of where on that explanatory variable you started or what the values of the ...


3

The probability of the outcome occurring at one specific opportunity never changes. However, the probability that the outcome occurs within a set of opportunities does change based on the size of that set. Coin flip example: What's the probability this flip will be "heads"? 50% What's the probability that I get at least one "heads", if I flip the coin ...


3

To answer your question consider a binary outcome Y (0 or 1) and a binary exposure variable X (0 or 1). In prospective studies we are interested in $P[Y = 1 | X = 0]$ v. $P[Y = 1 | X = 1]$ whereas in retrospective studies we are interested in $P[X = 1 | Y = 0]$ v. $P[X = 1 | Y = 1]$. First let's consider a prospective study with the setting below: $P_0 = P(...


3

This probability tree represents the game and guides the calculations: The blue node at the left represents the start. At this point there is a 5% chance of success (leading to the up and left). If success is achieved now, only one attempt is made, as indicated in the orange circle. Lacking success, we progress down and to the right to the next blue node....


3

Remember that only the coefficient for the constant from logitsitc regression are log odds; the remaining coefficients are log odds ratios. Lets say your baseline odds ($\exp(constant)$) is .8. This would mean that there are 0.8 people who say yes for every person who says no when all explanatory variables are 0. Your odds ratio for A says that a unit ...


3

You need to formulate a sequence of conditional probabilities which can then be multiplied by each other. Single flight survival Pr(S) = 1=Pr(crash). So id p= Pr(crash), the probability of surviving all of 1000 flights is the probability of surviving a single flight raised to the 1000th power = (1-p)^1000 ...or with R: (And think of the frequent flyer miles ...


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