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For ordered response categories, model each displayed conditional probability by a grouped-binomial logistic regression. The multinomial likelihood factors exactly into successive conditional binomial likelihoods, using the remaining category total as each stage's denominator. This is a different regression parameterization from baseline-category logits or cumulative odds. Separate cumulative binary indicators from the same person are dependent and cannot be treated as independent likelihood contributions.

Assume a constant annual force of infection $\lambda >0$ across ages and calendar years. For independent sampled people with ages $a_i>0$ and binary results $y_i$, fitApplying the complementary log-log link givesTherefore fit a binomial regression with the complementary log-log link, an intercept, and $\ln a_i$ as an offset with coefficient fixed at one. Age is measured in years. An ordinary freely fitted coefficient of age, or even a freely fitted coefficient of $\ln a_i$, would describe a different exposure model. The annual force of infection estimate isAn infection-naive person's one-year infection probability is $1-e^{-\hat{\lambda }}$, which is approximately $\hat{\lambda }$ only when that rate is small.

The log-likelihood being maximized is, apart from constants,A confidence interval for ${\beta }_0$ can be exponentiated to give a positive interval for $\lambda$. For groups of $n_j$ people at a common age $a_j$, use $Y_j\sim \mathrm{Binomial}(n_j,1-e^{-\lambda a_j})$ with the same offset. If an age band is broad, use individual ages or average $1-e^{-\lambda a}$ over the band's age distribution instead of treating its midpoint as exact. No numerical estimate is possible without the survey observations.

The biological interpretation assumes that immunoglobulin G positivity records previous infection, persists over the relevant ages, and is not substantially produced by maternal antibodies or vaccination. These are conditions to check, rather than conclusions from the fitted curve. Use known-positive and known-negative serum controls to assess the assay threshold, and validate specificity against relevant other infections. Cross-reactive antibodies to other flaviviruses can produce positive dengue virus serology, so a positive result need not be a marker specific to dengue virus exposure.^[ref]

If independently established assay diagnostic sensitivity and diagnostic specificity are $Se$ and $Sp$, the observed positive probability isUse that observation model in the binomial likelihood; it is generally no longer the simple intercept-plus-offset complementary log-log link fit. With only the stated survey, one cannot freely estimate $Se$, $Sp$, antibody loss and an arbitrary infection history simultaneously. A single binary result also does not identify how many subsequent infections a person has experienced.

For a group with sample size $n_s$, let $Y_s$ be its overweight count. The natural sampling model is a binomial distribution, $Y_s\sim \mathrm{Bin}(n_s,{\pi }_s)$, with observed proportion $p_s=Y_s/n_s$. A grouped-binomial logistic regression specifiesFitting event/nonevent counts, or fitting the proportions with weights $n_s$, gives the same binomial likelihood. The weights represent trial counts; unweighted regression of the proportions would give small and large groups the same influence. If printed percentages are rounded, exact original event counts are preferable to manufacturing fractional counts from $n_s{p}_s$.

Year should ordinarily be a factor with three levels unless a linear trend over time is specifically intended. Country, sex and age group are also possible factors. The actual subset, reference levels and formula are absent from this PDF. For a year-by-age interaction term, writeWith treatment contrasts, $\alpha$ is the baseline logit, $u_a$ and $v_t$ are effects at the reference level of the interacting factor, and $w_{at}$ is an additional difference of log-odds differences. Exponentiating a main-effect contrast gives an odds ratio; exponentiating an interaction contrast gives a ratio of odds ratios. It is incorrect to interpret a main effect as a constant effect across the other factor when the interaction term is present.

A coefficient table should be read as estimate, standard error, a Wald estimate/error ratio and its reference-distribution probability. A whole multi-level interaction term is better tested by a nested likelihood-ratio test, using the decrease in deviance and the number of added independent coefficients. Null and residual deviances, residual degrees of freedom, convergence and fitted probabilities should all be checked. Large residual dispersion suggests lack of fit or dependence within sampled schools; a quasibinomial regression or an appropriate clustered analysis may then be preferable. Residual deviance is not automatically a valid chi-squared goodness-of-fit statistic for sparse groups or after estimating a free dispersion.

An interaction plot should display the group means or fitted probabilities against year, with separate lines for the interacting factor, and should also inspect the link scale. Nonparallel probability curves alone do not establish a logit interaction: even an additive logit model transforms nonlinearly to probabilities. As an independent descriptive illustration, the original study's Table 2 reports English boys' age-specific percentages. The following original plot shows their widening age differences after 1984 on both scales. It uses published descriptive values, not a reconstruction of the absent examination fit. The data are from pmc.ncbi.nlm.nih.gov/articles/PMC26603/ .

For example, on the probability scale the 1984–1994 increase for the oldest group exceeds that for the youngest by $6.9-0.8=6.1$ percentage points using the rounded table values. On the logit scale the corresponding difference of differences isThis describes the interaction pattern, but its sampling uncertainty requires the group counts. The absent output prevents identifying the exam fit's coefficient estimates and significance tests.

There are two different uses of Poisson regression. A rare-event approximation models $Y_s\sim \mathrm{Pois}(n_s{\pi }_s)$ withThe fixed offset $\log n_s$ accounts for exposure, and exponentiated slopes are risk ratios, not odds ratios. The approximation is most appropriate when probabilities are small; it is not exact for appreciable prevalence because binomial variance is $n_s{\pi }_s(1-{\pi }_s)$ rather than $n_s{\pi }_s$.

An exact alternative stacks event and nonevent counts in each group and fits a log-linear model with a separate group intercept and event-by-predictor effects:Conditioning on their total $n_s$ gives the binomial event probability ${\mu }_{s1}/({\mu }_{s0}+{\mu }_{s1})$, whose logit is exactly $x_s^T\beta$. Profiling the group intercepts enforces the observed totals and reproduces the grouped-binomial logistic regression. This Poisson surrogate for a conditional multinomial model is exact and should not be confused with the rare-event approximation.

A direct grouped-binomial logistic regression can dichotomize satisfaction, for example high versus low-or-medium. In each covariate stratum use the high count as successes and the sum of the other counts as failures, with total $N_{ijk}$. This models $P(Y=\mathrm{high}\mid i,j,k)$ and is appropriate if that binary contrast is substantively intended, but discards distinctions among the merged categories. Another cutpoint gives a different binary question.

There is also an exact sequential binomial factorization preserving all three categories. With $p_1,p_2,p_3$ their probabilities in one stratum,Multiplying these two conditional binomial likelihoods gives the multinomial likelihood: the combinatorial coefficients multiply to $N!/(n_1!n_2!n_3!)$, and the probability factors reduce to $p_1^{n_1}{p}_2^{n_2}{p}_3^{n_3}$. Continuation-ratio logits can therefore fit two logistic regressions, using the remaining stratum total as the second denominator. These are conditional stage models, not two independent counts with the same fixed denominator. Their regression restrictions need not equal those of the additive baseline-category model.

Finally, an ordinal analysis may fit cumulative binary contrasts and examine whether their slopes can be shared in a proportional-odds model. The binary indicators at different cutpoints from the same person are dependent. Do not stack them as independent binomial observations and multiply their marginal likelihoods; fit the joint ordinal likelihood or use a method with the appropriate dependence correction. This distinction separates a convenient descriptive collection of binary fits from a coherent multinomial or ordinal model.

At a distinct event time $a_m$, estimate the conditional probability of surviving that time by $1-d_m/r_m$. Multiplication of these conditional survival probabilities gives the Kaplan–Meier estimator, and taking minus its logarithm gives an integrated hazard estimate:The conditional failure count has a binomial likelihood whose maximizing event fraction is $d_m/r_m$. Tied failures therefore enter directly as one count. Even ordering $d_m$ failures arbitrarily, with no intervening censoring, produces the same survivor function factor becauseThe failure-versus-censoring ordering convention still matters when both have the same recorded time. If $d_m=r_m$, the survivor function becomes zero and the logarithmic integrated hazard is infinite.

Let $p_1$ be the alternative death probability. The odds ratio specifiesAssume $0<p_0<1$ and $k>0$. For observed batch count $y_t$, the common combinatorial factor in the two binomial likelihoods cancels, giving the CUSUM incrementThereforeFor $k>1$ large death counts add positive evidence of deterioration; for $k<1$ unusually small death counts add evidence for improvement. The score has nonpositive expectation under the null and nonnegative expectation under the specified alternative, as follows from the nonnegativity of Kullback-Leibler divergence. Independent, nonoverlapping batches avoid counting the same patient's outcome repeatedly.

Put $N=n_0+n_1+n_2$ and $s=2x=2\theta (1-\theta )$. The usual recombination fraction range $0\le \theta \le 1/2$ gives $0\le s\le 1/2$, and the preceding likelihood function becomesTo see its relation to nonparametric linkage analysis, use Bayes' theorem and the constant disease probability $1/16$ to obtain the affected-pair marker-sharing probabilitiesTheir multinomial likelihood has exactly the same $s$-dependent kernel, since its extra factor $2^{n_1}$ is constant. Set $A=2n_0+n_1$ and $B=2n_2+n_1$. Then $A+B=2N$, and $B$ is the total number of shared ancestral copies. At fixed $N$, the Fisher-Neyman factorization theorem makes $B$ a sufficient statistic. The observed mean shared proportion is $\hat{p}=B/(2N)$, and the binomial likelihood kernel givesThe unlinked null is $s=1/2$, whereas linkage corresponds to $s<1/2$, or expected shared proportion $1-s>1/2$. If $\hat{p}\le 1/2$, the constrained likelihood is maximized at the unlinked null. For $\hat{p}>1/2$, its log likelihood ratio iswhose derivative in $\hat{p}$ is $2N\log [\hat{p}/(1-\hat{p})]>0$. Hence it orders samples exactly by excess IBD sharing. Under this fully penetrant recessive model, parametric linkage testing and a one-sided test of the overall IBD-sharing proportion use the same information. This is allele-sharing sufficiency for recessive linkage, not a general equivalence for all disease models. Use limiting values for terms such as $0\log 0$, and assume $N>0$ for the estimators.

The law of total probability and the definition of conditional probability give Bayes' theorem:The denominator is positive because $P(E)>0$.

For the observed cargo, set $p_A=0.05$, $p_B=0.03$, $p_C=0.01$. The binomial likelihood under origin $i$ isEqual prior probabilities and Bayes' theorem make the posterior proportional to these likelihoods. The binomial coefficient cancels in the likelihood ratios:Thus $L_A/L_B\approx 4.99\times 10^{-45}$ and $L_C/L_B\approx 2.74\times 10^{-9}$, givingEven the permitted approximation $\log (1-p)\approx -p$ gives log ratios about $-93.83$ and $-23.72$, which already show overwhelming posterior preference for B. The exact logarithms are preferable because multiplication by $9800$ magnifies the approximation error. An observed contamination fraction halfway between two hypothesized rates does not give equal likelihoods: the binomial distribution assigns asymmetric probabilities to those deviations, especially at this large sample size.

The binomial likelihood is $p(x\mid \theta )=\left(\genfrac{}{}{0ex}{}{m}{x}\right){\theta }^x(1-\theta {)}^{m-x}$. Multiplication by the beta distribution prior density and Bayes' theorem give the posterior density kernel ${\theta }^{a+x-1}(1-\theta {)}^{b+m-x-1}$. Normalizing by the beta function yields Beta-binomial conjugacy:This is proper for $a,b>0$ and $0\le x\le m$, including counts at either boundary.

The first fit is grouped-binomial logistic regression: independently, $Y_i\sim \mathrm{Bin}(N_i,{\pi }_i)$, with the observed group sizes $N_i$ andThe reference group is under 50 with a nonmalignant initial tumour. The weights specify binomial numbers of trials, not six equally weighted individual observations. There is no age-by-malignancy interaction. Each age coefficient compares that age group with the reference age at fixed malignancy; $\exp {\beta }_3$ is the common malignancy odds ratio.

The maximum-likelihood estimates are obtained by maximizing the binomial likelihood. The reported standard errors come from the inverse fitted Fisher information. Each displayed Wald test tests a coefficient equal to zero, using $z=\hat{\beta }/\mathrm{se}(\hat{\beta })$ and the approximate standard normal distribution; its two-sided probability is $2\Phi (-|z|)$. At the 5% level the middle age coefficient and the malignancy coefficient are significant, with negative signs; the oldest age coefficient is not, although its 9.17% probability is weaker evidence in the same direction. The oldest group has only 19 observations and its larger standard error does not establish absence of an age effect. The intercept tests reference log-odds zero, equivalently reference survival probability $1/2$, rather than an age contrast. The reference fitted probability is about $0.888$; the fitted malignancy odds ratio is about $0.481$.

The binomial deviance is twice the saturated-minus-fitted log-likelihood. The null fit has one parameter, hence $6-1=5$ residual degrees of freedom; the first fit has four parameters, hence $6-4=2$. The improvement $12.65585-0.49409=12.16176$ is an approximate likelihood-ratio test on three degrees of freedom, with probability about $0.0069$. The residual deviance $0.49409$, compared with ${\chi }_2^2$, gives about $0.781$, so there is no evidence of lack of fit. These are asymptotic comparisons, especially approximate for the small oldest strata. The Akaike information criterion is $-2\ell +2p$, here with $p=4$.

Pooling the two older age groups gives a grouped-binomial logistic regression with a common coefficient for age at least 50 and three parameters in total. The similar older-age estimates and the sparse oldest group motivate examining this restriction, but the significance of either coefficient separately is not a test of their equality. The actual analysis of deviance for pooling compares the nested fits: the deviance increases by only $0.784-0.49409\simeq 0.28991$ for one lost parameter, giving a ${\chi }_1^2$ probability about $0.590$. The pooled model also has smaller Akaike information criterion, $28.723$ rather than $30.433$. Prefer the pooled model for its simpler adequate fit. Its common age and malignancy coefficients are negative and significant at 5% in the reported Wald tests.

There is a literal parenthesis error in the printed final R fragment: the first sum is not closed before subtracting the second sum. As printed, the expression is incomplete. Closing that sum gives twice the saturated-minus-fitted log-likelihood, including the binomial constants which cancel. Its intended final value is therefore the second fit's binomial deviance, $\boxed{0.784\text{ approximately}}$.

Zero observed events do not mean that the underlying event probability cannot be estimated. For independent observations having a common Bernoulli distribution, the binomial likelihood with no events is $(1-p{)}^{1515}$, maximized at $\hat{p}=0$. A zero-event binomial upper confidence bound shows the remaining uncertainty: the one-sided 95% upper bound solves $(1-p_U{)}^{1515}=0.05$, givingThus the point estimate is zero, but a nonzero mortality risk is compatible with the observations. This bound is an illustration under independent equal-risk sampling; clustered data or unequal risks require their own uncertainty calculation.

The empirical unadjusted odds ratio is also defined:Its log odds ratio is $-\infty$, so the ordinary normal approximation for a log-odds confidence interval is unavailable. Exact likelihood-based limits can still describe uncertainty; a specified continuity correction would instead yield a finite, method-dependent point estimate.

In an unpenalized logistic regression with a separate coefficient for elective exposure, every exposed death outcome is zero. Decreasing that coefficient toward $-\infty$ increases the exposed observations' likelihood contributions and leaves the reference observations unchanged. This is quasi-complete separation: there is no finite ordinary maximum-likelihood estimate of the exposure coefficient, even after adding background predictors. It is reasonable to say that the usual finite adjusted coefficient could not be fitted; it is too strong to say that no risk estimate or statistical information is possible. A specified penalized likelihood or Bayesian inference with proper coefficient priors can provide a finite adjusted estimate, whose dependence on the regularization must be reported.

Index independent experimental vials by $r$, write $N_r$ for the observed total eaten and $E_r$ for the number of other-species eggs eaten. The grouped-binomial logistic regression usesHere $D_r\in \{1,2,3\}$ and $S_r\in \{A,B,C\}$ identify day and adult species, and the free effects are ${\beta }_0,d_2,d_3,s_B,s_C$. There is no day-species interaction. The weights in the binomial generalized linear model are the denominators $N_r$, so fitting the proportions $E_r/N_r$ gives this binomial likelihood, not 23 Bernoulli observations. The dispersion is fixed at one. This is a model for the composition of eggs eaten conditional on the total, rather than a model for the total appetite.

Compare the species-only model against the additive day-and-species model. The null hypothesis is $H_0:d_2=d_3=0$, with the alternative that at least one is nonzero. The likelihood-ratio statistic is the reduction in residual deviance,Under the null hypothesis, regular binomial likelihood asymptotics give $G^2\stackrel{\mathrm{approx}}{\sim }{\chi }_2^2$, because the larger model has two additional parameters. For this chi-squared distribution the upper-tail probability is $\exp (-0.341/2)\simeq 0.843$. There is no evidence of a day effect after controlling for species, and the species-only model is a reasonable simpler choice. The sequential day-only statistic 0.273 from the preceding table answers a different question and must not replace this comparison.

With the uniform prior, the binomial likelihood gives the beta distribution posterior $\mathrm{Beta}(s+1,n-s+1)$. Under squared-error loss, the Bayes estimator is the posterior mean:

The production manager's prior gives posterior density proportional to $p^{s+1}(1-p{)}^{n-s}$. Minimizing $\mathbb{E}[(1-p)(\hat{p}-p{)}^2\mid X=s]$ gives the weighted posterior meanusing the supplied beta function. Cross multiplication shows ${\hat{p}}_P>{\hat{p}}_G$ exactly when $n>2s$, so it is greater unless $s\ge n/2$.

Ignoring factors that do not depend on the parameters, the product of the two binomial likelihoods isIncluding the sampling constants multiplies this by $\left(\genfrac{}{}{0ex}{}{200}{120}\right)\left(\genfrac{}{}{0ex}{}{220}{110}\right)$.

Assume each sampled response is truthful, card selection is independent of the student's status, and responses follow the usual independent sampling model. In randomized response, a yes answer occurs either for a user receiving the first question or for a nonuser receiving the complementary question. Its probability isConsequently $Y\sim \mathrm{Bin}(n,q)$, and the likelihood isFor $x\ne 1/2$, first maximize the binomial likelihood in the unrestricted probability $q$: its maximum occurs at $Y/n$. Inverting the affine relation givesFor an interior value $0<Y<n$, the same result follows from the scoreThe endpoint cases follow by monotonicity. This estimator has expected value $p$ and varianceThese identities follow immediately from $\mathbb{E}Y=nq$ and $\mathrm{Var}(Y)=nq(1-q)$.

The printed claim needs a parameter-space qualification. The admissible yes probabilities lie in $[\min (x,1-x),\max (x,1-x)]$, which may exclude $Y/n$. The constrained maximum-likelihood estimator isThe displayed untrimmed estimator is the maximum-likelihood estimator only when it already lies in $[0,1]$. Its unbiased estimator property does not carry over to the clipped estimator in general. For example, if $n=1$, $x=3/4$ and $p=0$, the clipped estimate equals 1 exactly when the answer is yes, so its expectation is $1/4$, not zero. At $x=1/2$, $q=1/2$ for every $p$: the parameter is unidentifiable and the printed quotient is undefined. Thus the unbiased formula requires $x\ne 1/2$, and the constrained likelihood assertion requires the stated clipping.

For any fixed $p>1/4$ below 1, the ratio to the null isIt increases with $y$. Its ordering is therefore the same for every such alternative, while the size is determined only by the null. The very same rule, reject when $Y\ge 4$, is most powerful against each alternative and hence is a uniformly most powerful test. At $p=1$ its power is 1, also optimal.

More generally, for $0<p_0<1$, the binomial likelihood ratio against any $p>p_0$ increases with $y$. Choose an integer cutoff $c$ so thatand reject for $Y>c$, randomize with probabilitywhen $Y=c$, and accept below. This gives exact size $\alpha$ and is a uniformly most powerful test for every positive integer $n$. Exact prescribed size may require this boundary randomization; a deterministic cutoff alone need not attain it. For $p_0=0$, reject all $Y>0$ and reject at $Y=0$ with probability $\alpha$, which is likewise optimal. At $p_0=1$ the stated one-sided alternative is empty. The cutoff generally depends on $p_0,n,\alpha$; there is no claim that one fixed cutoff serves every null parameter.

A psychometric function specifies the probability of a measured response as a physical stimulus variable changes. The response might be “present,” “clockwise,” or correct in a two-alternative forced-choice task. These are different dependent variables. Understanding the function's shape is necessary to distinguish sensory changes from decision, attention and measurement effects. A reported threshold without its task, response definition and performance criterion is incomplete. For example, a visual experiment could use calibrated display visual contrasts, timed two-interval presentations, fixation control and button choices; the observer’s choice supplies the dependent variable.

A smooth curve need not mean that a receptor has a smoothly varying conscious threshold. Consider a noisy internal response $R=\mu (x)+\epsilon$, with “yes” reported when $R>c$. Here $c$ is a decision criterion, not necessarily a biological detection limit. If $F_{\epsilon }$ is the noise cumulative distribution function,With Gaussian noise of standard deviation $\sigma$, symmetry of the normal distribution gives a Gaussian psychometric functionThe sigmoid is therefore a distribution tail swept by a changing mean response. At low means, few responses exceed the criterion; near it, many trials switch category; at high means, almost all do. The derivative is $\varphi ((\mu -c)/\sigma ){\mu }^{\prime }(x)/\sigma$, so slope depends on both sensory gain and variability. Moving $c$ shifts the curve without necessarily improving sensitivity. If noise vanished and the mean crossed the criterion sharply, this simple model would approach a step rather than a graded transition.

There is no universal sigmoid in every stimulus coordinate. A nonlinear transduction function changes the curve, and a logarithmic intensity axis changes its appearance. As a counterexample, a toy detector responding whenever at least one event occurs in a Poisson distribution with mean $kx$ has $P(\text{yes}\mid x)=1-e^{-kx}$. This is concave on a linear positive-intensity axis. It follows immediately by subtracting the zero-event probability from one. It is a mathematical example, not evidence that human detection universally uses a one-event rule. Variation in attention or decision criterion across trials can broaden a measured curve even if the within-state sensory noise is unchanged.

Task conventions determine the endpoints. A useful psychometric function with guessing and lapses isHere $\gamma$ is the lower asymptote and $\lambda$ the upper error asymptote. In balanced two-alternative forced-choice performance, $\gamma$ is commonly fixed at $1/2$. A yes/no response probability does not automatically have this lower bound. In this particular cumulative-Gaussian parameterization, $p(\alpha )=\gamma +(1-\gamma -\lambda )/2$; thus $\alpha$ gives 75% correct only for $\gamma =1/2$ and $\lambda =0$. Other function families use different scale parameters. Nor is $\lambda$ automatically the fraction of inattentive trials: if a fraction $r$ of trials involves random two-choice guessing, those trials contribute an upper error probability $r/2$.

The relation to signal detection theory is equally important. Under independent equal-variance Gaussian signal and noise samples, define sensitivity index $d^{\prime }=({\mu }_s-{\mu }_n)/\sigma$. In an ideal two-interval comparison, the response difference has mean ${\mu }_s-{\mu }_n$ and variance $2{\sigma }^2$, givingThis Gaussian forced-choice sensitivity relation follows by standardizing the Gaussian difference. It is a specified decision model, not an identity for every forced-choice experiment. In a yes/no task, hits and false alarms are both needed: under the corresponding equal-variance model, $d^{\prime }={\Phi }^{-1}(H)-{\Phi }^{-1}(F)$. A criterion change can increase both rates while leaving $d^{\prime }$ unchanged. The phenomenological fit above need not have parameters identical to these internal-response parameters.

These distinctions affect experiments. Felix A. Wichmann and N. Jeremy Hill used simulation and statistical modeling to show that unmodeled stimulus-independent errors can bias threshold and slope estimates, even when lapses are uncommon.^[ref] Ignoring high-intensity errors can make a function appear shallower and falsely suggest worse sensory precision. A broad group-average curve can likewise reflect a mixture of individually sharp functions with different locations.

At predetermined stimulus levels, independent trials give counts $k_i\sim \mathrm{Binomial}(n_i,p(x_i))$. The parameter-dependent log-likelihood is ${\sum }_i[k_i\log p_i+(n_i-k_i)\log (1-p_i)]$. Fitting this binomial likelihood, checking goodness of fit and quantifying uncertainty are preferable to drawing a curve through noisy proportions. Easy trials help constrain lapses; levels near the transition constrain location and slope. Randomization, stable calibration and checks for learning or fatigue help distinguish a stationary sensory relation from drift. Adaptive procedures require inference appropriate to their sampling design.

Thus the function is both a description and a testable model. Its shape determines what a threshold means, how an experiment should sample, and which claims about sensory mechanisms are warranted.

The prior mean, posterior mean, and maximum-likelihood estimate areFor $0<S<N$, the last follows by maximizing $S\log \theta +(N-S)\log (1-\theta )$. At $S=0$ or $S=N$, use the usual binomial likelihood convention that admits the endpoints $0$ and $1$ as maximum-likelihood estimates; if the parameter domain is strictly $(0,1)$, these are boundary suprema rather than attained maxima. The posterior mean remains interior in all cases.

TakinggivesSince $N=nm>0$ and $\beta +1>0$, $0<\lambda <1$. Thus the posterior mean is a convex combination of the prior mean and observed binomial proportion, with larger prior weight when $\beta +1$ is large relative to $N$.

Randomized response masks which of two complementary binary questions a person answered. Suppose the sensitive property has probability $p$, and independently the direct question is selected with known probability $x$, while the complementary question is selected otherwise. Truthful answers then have yes probability $q=xp+(1-x)(1-p)$. The investigator observes only the answer, preserving uncertainty about an individual's status while permitting population statistical estimation when $x\ne 1/2$.

For $n$ independent responses, $Y\sim \mathrm{Bin}(n,q)$, and the affine estimator $\tilde{p}=(Y/n-1+x)/(2x-1)$ is an unbiased estimator. Its variance is $q(1-q)/[n(2x-1{)}^2]$, by the mean and variance of a binomial distribution. Equivalently the variance is $p(1-p)/n+x(1-x)/[n(2x-1{)}^2]$, exposing the extra sampling uncertainty caused by the masking. At $x=1/2$, all $p$ produce $q=1/2$ and identifiability fails. At $x=0$ or $1$, there is no masking and the extra variance vanishes.

Because $0\le p\le 1$, maximum likelihood estimation gives $\hat{p}=\min (1,\max (0,\tilde{p}))$. This follows by maximizing the binomial likelihood over the interval $q\in [\min (x,1-x),\max (x,1-x)]$ and then inverting its monotone affine parametrization. Clipping can introduce bias: for $n=1$, $x=3/4$ and $p=0$, the clipped estimate has mean $1/4$.