Bayesian Hypothesis Testing for Hospitality and Tourism Research

Abstract

In hospitality and tourism research, p-values continue to be the most common approach to hypothesis testing. In this article, we elaborate on some of the misconceptions associated with p-values. We discuss the advantages of the Bayesian approach and provide several important practical recommendations and considerations for Bayesian hypothesis testing. With the main challenge of Bayesian hypothesis testing being the sensitivity of the results to prior distributions, we present in this article several priors that can be used for that purpose and illustrate their performance in a regression context.

Keywords

Bayesian hypothesis testing Bayes factor marginal posterior

Introduction

In hospitality and tourism research, and other related fields such as marketing and management, it is common to use p-values to test research claims or hypotheses. For the most part, p-values have been the sole criteria used to support or reject the hypotheses of interest. The American Statistical Association has recently issued a statement regarding a number of logical and practical limitations associated with p-values. The statement was mainly geared toward the misinterpretation and abuse of p-values, emphasizing the fact that “p-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone,” and that “scientific conclusions and business or policy decisions should not be based only on whether a p-value passes a specific threshold” (Wasserstein & Lazar, 2016, p. 131).

Importantly, p-values “do not measure the size of an effect or the importance of a result” and “do not provide a good measure of evidence regarding a model or hypothesis.” In a recent article, Assaf and Tsionas (2018) have discussed all these issues in detail and have provided simulation evidence about the “dancing p-values” phenomena, illustrating specifically how p-values can fluctuate between various samples testing the same hypothesis. The authors emphasized the importance of using Bayes factors (BF) or marginal posterior densities as an attractive alternative to p-values. The Bayesian approach, which is gaining popularity across marketing and management (Zyphur & Oswald, 2015), allows researchers to provide quantitative evidence for any hypothesis, including the null hypothesis. Importantly, the Bayesian approach relies only on the data at hand and not on some imaginary repeated samples that are not usually observed.

The goal of this note is to expand on the issues raised by Assaf and Tsionas (2018), further emphasizing the advantages of using the Bayesian approach for hypothesis testing and model comparison in hospitality and tourism research. We provide several important practical recommendations for future research.

Bayesian Hypothesis Testing

Background

In this note, we do not intend to provide a detailed background of the Bayesian approach, as this has been explained in detail elsewhere (see Assaf & Tsionas, 2018; Zyphur & Oswald, 2015). Generally, the Bayesian analysis proceeds as follows. Suppose we have a model with parameters $θ,$ likelihood function $L (θ; y),$ and prior $p (θ) .$ The available data is denoted by $y$ . Elementary operations yield Bayes’s theorem:

p (θ | y) = \frac{p (θ, y)}{p (y)} = \frac{p (y | θ) p (θ)}{p (y)} \equiv \frac{L (θ; y) p (θ)}{p (y)} .

The denominator is also $p (y) = \int L (θ; y) p (θ) d θ .$ It is the marginal probability to observe the data and it is known as marginal likelihood or “evidence.” Its role in Bayesian model comparison and testing can hardly be overemphasized.

Suppose now that we have two models, say A and B, with parameters $θ_{A}$ and $θ_{B},$ likelihood functions $L_{A} (θ_{A}; y)$ and $L_{B} (θ_{B}; y),$ and priors $p_{A} (θ_{A})$ and $p_{B} (θ_{B}) .$ To compare the models, we can use the posterior odds ratio in favor of Model A and against Model B:

P O R_{A B} = \frac{p_{A} (y)}{p_{B} (y)} \cdot \frac{p_{A}}{p_{B}},

(1)

where $p_{j}$ is the prior probability in favor of model $j$ ( $j = A, B) .$ The ratio $\frac{p_{A} (y)}{p_{B} (y)}$ is known as the BF, and $\frac{p_{A}}{p_{B}}$ is the prior odds ratio. If prior odds are the same, then the posterior odds ratio (POR) is equal to the BF. Often, $p_{j} (y)$ is denoted by $M_{j} (y) .$ Therefore, the BF is:

B F = \frac{M_{A} (y)}{M_{B} (y)} .

(2)

In the case of two parameters, viz. $θ = [θ_{A}, θ_{B}]$ the marginal posterior density of $θ_{A}$ is defined as:

p (θ_{A} | y) = \int p (θ_{A}, θ_{B} | y) d θ_{B} .

For a Bayesian, one of the most critical parts in hypothesis testing is to look directly at the marginal posterior densities of the parameters of interest. Suppose we have a general model whose parameters are $θ \in Θ \subseteq ℝ^{m},$ the likelihood function is $L (θ; y)$ and the prior is $p (θ) .$ By Bayes’s theorem, we have: $p (θ | y) \propto L (θ; y) p (θ) .$ The marginal likelihood is the integrating constant of the posterior, that is:

M (y) = \int_{Θ} L (θ; y) p (θ) d θ .

(3)

In general, this multivariate integral is not available in closed form. To compute an approximation, notice that:

p (θ | y) = \frac{L (θ; y) p (θ)}{\int_{Θ} L (θ; y) p (θ) d θ} = \frac{L (θ; y) p (θ)}{M (Y)},

(4)

from which we obtain:

M (y) = \frac{L (θ; y) p (θ)}{p (θ | y)} .

(5)

The important feature is that this holds identically for all $θ \in Θ$ and is known as “candidate’s formula.” Therefore, it holds for $\bar{θ}$ which is, say, the posterior mean. Therefore:

M (y) = \frac{L (\bar{θ}; y) p (\bar{θ})}{p (\bar{θ} | y)} .

(6)

The numerator can be computed easily in certain instances, but the denominator is unknown. If we assume that $p (θ | y)$ is approximately normal with mean $\bar{θ}$ and covariance matrix $\bar{V},$ then $p (\bar{θ} | y) ≃ {(2 π)}^{- m / 2} | \bar{V} |^{- 1 / 2} .$ From this expression and (6), we have:

\log M (y) = \log L (\bar{θ}; y) + \log p (\bar{θ}) + \frac{m}{2} \log (2 π) + \frac{1}{2} \log | \bar{V} | .

(7)

In this formula, $\bar{θ}$ and $\bar{V}$ can be computed easily from Markov Chain Monte Carlo (MCMC) draws. In models with latent variables, the application of this formula is problematic unless the latent variables can be used to facilitate MCMC. However, it is usually easy to integrate them out and compute the log-likelihood in closed form. If this is not possible, then one can use importance sampling as in Perrakis et al. (2014).

Advantages and Challenges

As shown from Equation 2, for example, and in contrast to the p value approach, the Bayesian approach provides the researcher with several important advantages. First, the “Bayes factor is inherently comparative: It weighs the support for one model against that of another” (Andraszewicz et al., 2015, p. 529). It can be highly effective in the context of model comparison when the number of potential regressors is large and the objective is to select a parsimonious specification with good fit¹. Second, the BF deals only with the data at hand, and not with some imaginary samples that are not actually observed (as is the case with the frequentist approach). Third, the BF “provides evidence for and not only against $H_{0} .$ This is contrary to the (admittedly peculiar) habit of frequentists who do not reject a null but cannot accept it” (Assaf & Tsionas, 2018, p. 22). Fourth, the BF is better equipped to measure the size of an effect, and is a ratio of probabilities, and not a probability per se like the p value. Fifth, directly examining the marginal posterior density of a parameter has considerable advantages over the p value, particularly when the former is multimodal. The main challenge for Bayesian hypothesis testing is that the results can be sensitive to the prior distributions of the parameters reflecting the hypotheses one is testing. BFs are known to yield problematic results when diffuse or improper priors are used. In the remaining sections of this note, we aim to focus on this specific issue. We use a regression example to discuss several priors and illustrate their performance. We also provide important practical recommendations.

Regression Context

Suppose we have a regression model:

y_{t} = {x^{'}}_{t} β + u_{t}, t = 1, \dots, n,

(8)

or, in familiar notation $y = X β + u,$ where $x_{t} \in ℝ^{k}$ is a vector of explanatory variables, and $β \in ℝ^{k}$ is a vector of coefficients. The error terms are independent of the regressors, and $u \sim N_{n} (0, σ^{2} I_{n}) .$

Since $σ$ is rarely of interest, most researchers assume that the prior can be factored as $p (β, σ) = p (β | σ) p (σ),$ with $p (σ) \propto σ^{- 1} .$ Using a simple change of variables technique, it is easy to show that this corresponds to a prior for $\log σ$ that is flat along the real line, that is $p (\log σ) \propto const .$ Possible priors for $β$ are, of course, many. Two of the most prominent are:

β | σ \sim N_{k} (\bar{β}, σ^{2} \bar{V}),

(9)

that is, $β$ follows $a^{k}$ -variate normal distribution with mean vector $\bar{β},$ and covariance matrix $σ^{2} \bar{V},$ conditional on $σ,$ where $\bar{β}$ and $\bar{V}$ are to be specified by the user. An important alternative prior is:

β | σ \sim N_{k} (\bar{β}, \bar{V}) .

(10)

In this case, the prior of $β$ no longer depends on $σ,$ and $\bar{V}$ is the prior covariance matrix. Prior in Equation 9 is known as “conjugate” and allows one to analytically integrate the posterior with respect to $σ$ to obtain a closed form expression for the marginal posterior of $β,$ that is:

p (β | y) = \int_{0}^{\infty} p (β, σ | y) d σ .

(11)

Let us assume, in the interest of simplicity, the following “noninformative” or flat prior:

p (β, σ) \propto σ^{- 1},

(12)

which is the limiting case of Equation 10 when $\bar{β} = 0,$ and the elements of $\bar{V}$ diverge to infinity. Under Equation 10, we obtain the following explicit result about the marginal posterior of $β$ :

p (β | y) \propto {(ν + \frac{(β - b)^{'} X^{'} X (β - b)}{s^{2}})}^{- (ν + k) / 2},

(13)

where $ν = n - k,$ $ν s^{2} = (y - X b)^{'} (y - X b),$ $b = {(X^{'} X)}^{- 1} X^{'} y .$ Therefore, the marginal posterior distribution of $β$ is a k-variate Student’s-t with $ν$ degrees of freedom, location vector $b$ (the vector of OLS estimates), and scale matrix $s^{2} {(X^{'} X)}^{- 1} .$ This result nicely matches the sampling-theory result, in that the marginal posterior distribution of $β_{j}$ (for $j = 1, \dots, k)$ will be univariate Student’s-t with location $b_{j},$ and standard deviation given by $s {[{(X^{'} X)}^{- 1}]}_{i i} .$

With the more general prior (9) we have:

β | y \sim N_{k} (\hat{β}, V),

(14)

where

\hat{β} = {(X^{'} X + {\bar{V}}^{- 1})}^{- 1} (X^{'} y + {\bar{V}}^{- 1} \bar{β}),

(15)

and $V = s^{2} {(X^{'} X + \bar{V})}^{- 1} .$ Therefore, the marginal posterior distribution of $β$ is k-variate normal, with mean vector $\hat{β}$ and covariance matrix $V .$ The posterior mean ( $\hat{β}$ ) is interesting because when $\bar{β} = 0,$ and $V = h^{- 1} I_{k}$ for some constant $h > 0,$ then we obtain:

\hat{β} = {(X^{'} X + h I_{k})}^{- 1} X^{'} y,

(16)

which is the ridge-regression estimator. Interestingly, under the prior in Equation 10, it is no longer possible to integrate $σ$ out of the joint posterior $p (β, σ | y) .$ It is possible, nevertheless, to obtain the following posterior conditional distributions:

β | σ, y \sim N_{k} ({\hat{β}}^{*}, V^{*}),

(17)

where ${\hat{β}}^{*} = {(X^{'} X + σ^{2} {\bar{V}}^{- 1})}^{- 1} (X^{'} y + σ^{2} {\bar{V}}^{- 1} β),$ $V^{*} = σ^{2} {(X^{'} X + σ^{2} {\bar{V}}^{- 1})}^{- 1},$ and

\frac{(y - X β)^{'} (y - X β)}{σ^{2}} | β, y \sim χ_{n}^{2},

(18)

where $χ_{n}^{2}$ denotes the chi-square distribution with $n$ degrees of freedom. The posterior conditional distributions in Equations 17 and 18 can be used to implement a Gibbs sampler to provide draws ${β^{(d)}, σ^{(d)}, d = 1, \dots, S}$ that converge to the joint posterior distribution. As a matter of fact, Equation 18 suggests that if we adopt the prior:

\frac{\bar{q}}{σ^{2}} \sim χ_{\bar{n}}^{2},

(19)

where $\bar{n}, \bar{q} \geq 0$ are prior parameters, then the conditional posterior is:

\frac{\bar{q} + (y - X β)^{'} (y - X β)}{σ^{2}} | β, y \sim χ_{n + \bar{n}}^{2} .

(20)

Other Priors and Some Important Notes

As mentioned above, the main challenge for Bayesian hypothesis testing is that the results can be sensitive to the prior distributions of the parameters reflecting the hypotheses one is testing. Along with the priors in Equations 9 and 10, we present in this section other priors that can be used for Bayesian hypothesis testing.

Zellner and Siow (1980) Prior

Andraszewicz et al. (2015) recommended using the Jeffreys–Zellner–Siow priors as default priors for linear regression (Jeffreys, 1961, 1973; Liang et al., 2008; Zellner & Siow, 1980).

Zellner and Siow (1980) suggested the following prior:

p (β, σ) = p (β | σ) σ^{- 1},

(21)

and

p (β | σ) \propto {(1 + \frac{β^{'} X^{'} X β}{n σ^{2}})}^{- (k + 1) / 2} .

(22)

This is a $k$ variate Cauchy distribution², location is a zero vector, and scale matrix $σ^{2} {(X^{'} X)}^{- 1} .$ Based on this prior, in turn, they suggested approximate BFs for testing a variety of hypotheses about $β .$ Notice that this prior depends on $σ .$

To illustrate, suppose for example, we can write the regression model as:

y = X_{1} β_{1} + X_{2} β_{2} + u,

(23)

where $X_{1}$ is $n \times k_{1},$ $X_{2}$ is $n \times k_{2}$ $(k_{1} + k_{2} = k),$ and $β_{1}, β_{2}$ are, respectively, $k_{1} \times 1$ and $k_{2} \times 1$ . Often it is of interest to test the hypothesis:

H_{A} : β_{2} = 0,

(24)

compared with

H_{B} : β_{2} \neq 0,

(25)

in the sense that at least one element of $β_{2}$ is nonzero.

To see how the BF works, it makes sense to consider the following POR:

K_{A B} = \frac{\int \int p (β_{1}, 0, σ | y) d β_{1} d σ}{\int \int p (β, σ | y) d β d σ} .

(26)

The reader should notice that the POR is nothing but a ratio of two marginal likelihoods (and it is, therefore, a BF if the prior odds ratio is 1). The numerator imposes the restriction in $H_{A}$ while the denominator incorporates $H_{B}$ by “integrating out” uncertainty with respect to the entire parameter vector, consisting of $β$ and $σ .$

For “large” values of measure $K_{A B},$ known as Bayes Factor (when the prior odds are equal), one ought to accept the hypothesis, otherwise there is no evidence in favor of Equation 24. BF is the ratio of two “integrated likelihoods”; the first is evaluated under H₁ and the other leaves $β_{2}$ unspecified. It resembles the likelihood ratio statistic but instead of taking the maximum over $β_{2}, σ$ in the numerator, and $β, σ$ in the denominator, it averages the likelihood against the prior. To see this more clearly, notice that by Bayes’s theorem we have: $p (θ | y) = \frac{L (θ; y) p (θ)}{\int_{Θ} L (θ; y) p (θ) d θ} \propto L (θ; y) p (θ)$ , where $θ \in Θ \subseteq ℝ^{m}$ is a parameter vector, $p (θ)$ is the prior, and $L (θ; y)$ is the likelihood function (the joint distribution of observables given a value of the parameter vector $θ$ ). Using this result in Equation 26, we have:

K_{A B} = \frac{\int \int L (β_{1}, 0, σ; y) p (β_{1}, 0, σ) d β_{1} d σ}{\int \int L (β, σ; y) p (β, σ) d β d σ} .

(27)

Since $p (β, σ) = p (β | σ) p (σ),$ we have:

K_{A B} = \frac{\int \int L (β_{1}, 0, σ; y) p (β_{1}, 0 | σ) σ^{- 1} d β_{2} d σ}{\int \int L (β, σ; y) p (β | σ) σ^{- 1} d β d σ},

(28)

where $p (β_{1}, 0 | σ)$ is the conditional prior of $β$ evaluated at $β_{2} = 0,$ and we assume $p (σ) \propto σ^{- 1} .$ In this form, it is clear that the BF is, indeed, the ratio of two averaged likelihoods, where the weighting functions are the priors. For the normal regression model, using the Cauchy prior in Equation 22, it is possible to find an approximation of BF in closed form, as in Zellner and Siow (1980). By (3.13) in Zellner and Siow (1980) we have, specifically:

- 2 \log K_{A B} \approx ν_{B} \frac{R_{B}^{2} - R_{A}^{2}}{1 - R_{B}^{2}} - k_{2} \log ν_{B},

(29)

where $\approx means$ “approximately, in large samples,” $ν_{B} = n - k_{2},$ and $R_{A}^{2}, R_{B}^{2}$ are the coefficients of determinations under models $H_{A}$ and $H_{B},$ respectively. Of course, this approximation can be computed easily using commonly available software.

Importantly, we note that the recommendation of Andraszewicz et al. (2015) should not to be taken at face value. We examined this issue using an artificial data set with $n = 20$ and $k = 10$ regressors including the intercept. Regressor $x_{t 1}$ was generated as standard normal, and $x_{t j} = x_{t 1} + 0.1 ζ_{t j}$ ( $j = 2, \dots, 9$ ). All regression coefficients were equal to 1. As the regressors were correlated, we observed multicollinearity and most pairwise correlation coefficients were close to 0.9. We examined two cases, viz. $σ = 0.1$ and $σ = 1$ . We used 10,000 simulations. The results³ are reported in Figure 1. We focused on how the POR/BF performed.

Figure 1

Sample Distributions of Zellner–Siow (1980) $l o g K_{A B}$

When $σ = 0.1,$ the method performed well (Figure 1- panel (a)). However, when $σ = \frac{1}{2}$ the posterior odds in favor of $H_{A}$ were also “large” in the sense that $\log K_{A B} > 0$ (Figure 1, Panel b); although this should not be the case since $H_{A}$ says that the regression parameters are zero⁴. When $n = 1000$ (Figure 1, Panels c and d) clearly shows much better results and does not provide support for $H_{A} .$

We also report the marginal posterior densities of the parameters obtained under this prior (Figure 2). We apply MCMC techniques⁵ to provide access to these posteriors when $σ = 0.1$ and $n = 20,$ n = 100 $n = 100$ , and $n = 500$ . The results are reported in Panels a, b, and c of Figure 2 when $n = 20$ and in Panels d, e, and f when $n = 100$ . For $n = 500,$ the marginal posteriors are reported in Panels g, h, and i. All posteriors are for a single data set.

Figure 2

Marginal Posteriors of $β_{j}$ s and $σ$ With a Zellner and Siow (1980) Prior

Figure 3

Sample Distributions of $l o g K_{A B}$

As shown in Figure 2, some marginal posteriors (those corresponding to $β_{1}$ – $β_{5}$ ) have fat tails that are noticeable even when $n = 500$ . Some marginal posteriors (e.g., Panels a-c) are extremely fat–tailed when the sample size is quite small ( $n = 20$ ) and, therefore, the effect of using a Cauchy prior is much more pronounced.

Zellner’s g-Prior

Another prior that can be used is the Zellner’s (1986) g-prior, which rests on the ordinary least square (OLS) result that the covariance matrix of the OLS estimator is $V_{O L S} = s^{2} {(X^{'} X)}^{- 1} .$ Therefore, conditional on $X,$ it is reasonable to adopt the following prior:

β | σ \sim N_{k} (\bar{β}, g^{- 1} σ^{2} {(X^{'} X)}^{- 1}),

(30)

for some scalar $g > 0$ . It is not difficult to show that the marginal posterior of $β$ is:

β | y \sim N_{k} (\tilde{β}, \tilde{V}),

(31)

where

\tilde{β} = \frac{1}{1 + g} b + \frac{g}{1 + g} \bar{β},

(32)

and $\tilde{V} = \frac{σ^{2}}{1 + g} {(X^{'} X)}^{- 1}$ . Here, $b = {(X^{'} X)}^{- 1} X^{'} y$ is a vector of the OLS estimates. If $g = 0,$ we obtain the OLS estimate as the posterior mean. If $g \to \infty,$ the posterior and prior means are the same. In the general case, the posterior mean is a weighted average of the prior mean and the OLS estimates. The marginal likelihood of this model is available in closed form and can be easily evaluated. This has allowed some studies to perform model comparison with large numbers of variables, such as in growth regression studies (Fernandez et al., 2001). Provided the regressors have mean zero, the marginal likelihood is:

M (y) \equiv \int p (β, σ) d β d σ \propto {(\frac{g}{g + 1})}^{k^{'} / 2} {[\frac{1}{g + 1} y^{'} M y + \frac{g}{g + 1} T S S]}^{- (n - 1) / 2},

(33)

where $M = I_{n} - X {(X^{'} X)}^{- 1} X^{'}$ , and $T S S = {(y - \bar{y} 1_{n})}^{'} (y - \bar{y} 1_{n}) .$ The expression may be simplified to $M (y) \propto {(\frac{g}{g + 1})}^{k^{'} / 2} {[\frac{1}{g + 1} R S S + \frac{g}{g + 1} T S S]}^{- (n - 1) / 2},$ where $k^{'} = k - 1$ (viz. $k$ is the total number of parameters but $k^{'}$ is the number of regressors without the intercept, a convention that we adopt in this subsection), $R S S = {\hat{u}}^{'} \hat{u}$ , and $\hat{u} = y - X b$ are OLS residuals, and therefore $M (y) \propto g^{k^{'} / 2} {(g + 1)}^{- (n - k - 1) / 2} {(g + 1 - R^{2})}^{- (n - 1) / 2} .$ Fernandez et al. (2001) recommend the value $g = \frac{1}{m a x {n, k^{2}}} .$ For other selection rules for $g,$ see the literature summary in Fernandez et al. (2001).⁶

An Alternative Multivariate Cauchy Prior

Another option is the following multivariate Cauchy prior:

p (β) \propto {(1 + \frac{(β - \bar{β})^{'} (β - \bar{β})}{h^{2}})}^{- 1},

(34)

where the prior location parameter vector is $\bar{β} = 0$ and $h = 1$ . Compared with the Zellner and Siow (1980) prior, this differs in two respects: first, we do not condition on $σ$ and second, we do not condition on $X^{'} X$ (which was done in Zellner & Siow, 1980, for convenience).

Although we cannot recommend universal adoption of this prior, it does appear that it behaves fairly well in most situations.⁷ To validate, we again use the regression example in Section 4.1, which incidentally has a level of collinearity. We examined two cases: (a) $n = 20,$ $σ = 0.1,$ and (b) $n = 1000,$ $σ = 0.5$ . We used MCMC with 6,000 iterations (omitting the first 1,000) and considered 100 different data sets. The distribution of log BFs is shown in Panel a of Figure 4. There was considerable support in favor of the full model and against the hypothesis that the last five coefficients are zero. Notice that we have the same $H_{A}$ and $H_{B}$ as in Equations 24 and 25. For the computation of $K_{A B}$ we made use of Equation 7, that is, the Laplace approximation to the marginal likelihood of each model.

Figure 4

Sample Distributions of $l o g K_{A B}$

From the results in Panels a and b of Figure 4 (corresponding respectively to Cases a and b above), we saw that support for a is quite limited when $n = 20$ and $σ = 0.1$ and practically nonexistent in Case b where $n = 1000$ and $σ = 0.5$ . This is because in panel (b) of Figure 4, $\log K_{A B}$ ranges from −90 to about −30, which corresponds to values of $K_{A B}$ near $\exp (- 90)$ and $\exp (- 30)$ which are $8.2 \times 10^{- 40}$ and $9.4 \times 10^{- 14} .$ However, in Figure 4, Panel a, $\log K_{A B}$ can range from 0 to about 5, which corresponds to values of $K_{A B}$ from 1 to 148.14. So, the evidence in favor of the wrong model $H_{A}$ can be “strong” or “decisive” but is purely due to the small sample size and is included here only for purposes of illustration.

In general, the multivariate Cauchy prior we propose here is remarkably “robust” in the sense that posterior inferences are approximately the same, irrespective of extreme values for ${\bar{β}}_{j}$ s and $h$ (such as $- 10$ and $10^{- 12}$ in this data set).⁸

Practical Recommendations and Concluding Remarks

As presented in Andraszewicz et al. (2015, p. 540): “One may argue that in many situations, the data will pass the interocular traumatic test (i.e., when the pattern in the data is so evident that the conclusion hits you straight between the eyes; Edwards et al., 1963), and the results will be clear no matter what statistical paradigm is being used. Luckily, this is true; however, some data fail the interocular traumatic test and the results may indeed depend on the statistical paradigm that is used. In such cases, it seems worthwhile to not just base one’s statistical inference on frequentist methods alone but also rely on Bayesian techniques.”

We think one needs to be careful about this advice as it does not favor explicit articulation of one’s prior beliefs. Rarely, if ever, does one embark on research without prior knowledge. All papers cite previous literature, so, in practice, the researcher always has prior information. The question is how to incorporate such prior information into the analysis (a problem solved by Bayes’ theorem) and whether the posterior results are sensitive to reasonable deviations from the prior beliefs (that represent, e.g., beliefs that other researchers may reasonably entertain).

In this instance, looking at marginal posterior densities is quite informative and one is led to question the “validity” of prior information. Such priors are unlikely to holdup to the scrutiny of other researchers or they are unlikely to be shared by other researchers. Therefore, it is not necessary to look at both frequentist and Bayesian results; one can just look at Bayesian results and perform posterior sensitivity analysis with respect to the prior. One should also train themselves in looking at marginal posterior densities rather than relying on summary measures like the posterior mean and the posterior standard deviation. An often asked question is whether one can just look at the posterior mean and assign its ratio to the posterior standard deviation, the role of a “t statistic.” With a conjugate prior like Equation 10, one can certainly do so, as the marginal posterior of each $β_{j}$ is, in fact, a univariate Student’s-t distribution with $n - k$ degrees of freedom. Moreover, “when the sample size is large” one can interpret this ratio as a $z$ -statistic. Under a nonconjugate prior like Equation 9, the possible bimodality of marginal posteriors (see Panels c and d of Figure A.1-Appendix 2) makes this interpretation useless and one has to look at the entire marginal posteriors of the parameters of interest. Additionally, we know that as the sample size grows to infinity, marginal posteriors collapse to normal distributions with moments given by standard asymptotic theory.

The answer to the question of how large should a sample size be, really depends on the parameters of the model itself. For the toy model $y_{t} \overset{i i d}{\sim} N (μ, σ^{2}), t = 1, \dots, n,$ for example, with known $σ,$ we know that under a flat prior, $p (μ) \propto const .,$ the posterior of $μ$ is $N (\bar{y}, \frac{σ^{2}}{n}) .$ The sample is large enough, when it comes to estimating $μ$ by its posterior expectation, $\bar{y} = n^{- 1} \sum_{t = 1}^{n} y_{t},$ when $\frac{σ^{2}}{n}$ is small enough. What is “small enough” should be articulated by the researcher. To estimate $μ$ with precision 0.01, notice that we would need $2 \frac{σ}{\sqrt{n}} \leq 0.01,$ which implies $n \geq 40, 000 σ^{2} .$ If $σ = 0.1$ this implies $n \geq 400$ but when $σ = 1$ then the sample size should be huge. We presented above several important results from various priors. In general, we recommend the multivariate Cauchy prior with location 0 and scale h. In turn, researchers can elaborate more on the issue and articulate the location parameters (regression coefficients).

To sum up, and as Andraszewicz et al. (2015, p. 540) write: “So should management scientists become Bayesians? Given the lack of user-friendly software and course material.” The “lack of user-friendly software” is, of course, challenging; although programs like WinBUGS can be used to perform posterior analyses and prior sensitivity for most models used in management, marketing, and hospitality and tourism research. On the other hand, such lack is desirable as it discourages unwarranted reliance on “default priors” that may not be compatible with common sense beliefs about certain parameters. Some studies also recommend the comparison of p-values and the Bayesian approach to hypothesis testing. Again, we believe this recommendation is ill-advised. The two approaches measure quite different things and it is not easy to convince applied researchers that p-values and the Bayesian approach cannot be compared. For more sound advice, we refer the reader to Wasserstein and Lazar (2016).

Supplemental Material

Supplemental_material – Supplemental material for Bayesian Hypothesis Testing for Hospitality and Tourism Research

Supplemental material, Supplemental_material for Bayesian Hypothesis Testing for Hospitality and Tourism Research by A. George Assaf and Mike Tsionas in Journal of Hospitality & Tourism Research

Footnotes

ORCID iD

A. George Assaf

Supplemental Material

Supplemental material for this article is available online.

Notes

A. George Assaf (corresponding author; e-mail: assaf@isenberg.umass.edu) is a professor of Hospitality and Tourism Management, at the University of Massachusetts, Amherst. Mike Tsionas (e-mail: m.tsionas@lancaster.ac.uk) is a professor of Economics, Lancaster University Management School, the United Kingdom.

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