TREATMENT EXPECTATIONS AND THEIR IMPLICATIONS FOR LUMBAR FUSION SURGERY ON CHRONIC BACK PAIN

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TREATMENT EXPECTATIONS AND THEIR

IMPLICATIONS FOR LUMBAR FUSION SURGERY

ON CHRONIC BACK PAIN

Submitted by

Anna Warnqvist

A thesis submitted to the Department of Statistics in

partial fulfilment of the requirements for Master

degree in Statistics in the Faculty of Social Sciences

Supervisor

Adam Taube

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ABSTRACT

This study investigates the impact of expectations on treatment outcomes for patients with chronic lower back pain treated with Lumbar Fusion Surgery. Half the patients are randomly assigned to operation. Factors impacting reported outcome and satisfaction are investigated using beta regression and group LASSO. Treatment type is found to be the most important covariate affecting outcome, and outcome is found to be consistently related to satisfaction. In addition, the usability of beta regression on ordinal scales and visual analogue scales is inves-tigated. The method is found to perform well in presence of heteroskedasticity.

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1 Introduction - How did we end up here?

This thesis was to be an analysis on the impact of expectations on treatment outcomes of lumbar fusion surgery for patients with chronic back pain, and it is. But it is also a paper on statistical coping strategies when all reasonable outcome variables are problematic. The medical analysis is literally the backbone of this study, and we will always come back to it, but sometimes the sidetracks can be rather long.

The underlying cause of chronic back pain is often unknown, which makes the condition hard to treat [35]. Previous studies have investigated the effectiveness of lumbar fusion surgery on chronic back pain, but failed to find the operation to be effective [39, 13, 17, 7]. How-ever, the study from which data for this thesis comes from showed, that by following a careful diagnostics procedure, it is possible to identify a subgroup of patients to whom fusion oper-ation brings effective relief to pain. The data was already analysed in 2014 by Yinguan Zhu and evidence of a treatment effect was found [39]. The objective of this thesis is to analyse information on patient expectations concerning treatment forms and see how they reflect the outcome. The central question put to me by the medical professional in charge of the study was if overly positive expectations might cause patients to under-evaluate a reasonably good outcome. Investigating this is the primary goal of this paper.

Randomised controlled trials (RCT) involving surgery are rare and this study might be unique in investigating the expectations patients have of spinal surgery in a RCT setting. Expec-tations are of interest as they might help to control for discrepancies between clinical outcome and satisfaction, and the RCT setting provides a good framework to investigate this.

Other studies have worked with a similar problem setting.[9, 18] Fulfilment of treatment expectations has received more interest the past decade or so and there have been calls for using it as an end variable of interest alongside measurements of clinical outcome. Part of the reason for this change are the interest in patients’ general quality of life and related goals [38]. Other reasons are more financial; a satisfied patient will possibly be less likely to seek further care for the same condition, and thus save costs.[1] From this point of view the relationship between expectations and satisfaction is of great interest, and investigating it is the secondary goal of this paper.

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or no connection is found [1, 36, 9, 15]. There have also been results indicating that outcomes in pain level and function might not move to the same direction when their relationship to expectations are investigated [18].

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2 Data

The data consist of information from 74 patients with chronic back pain. Half of the patients (37) were randomly assigned to spinal fusion surgery. Both operated and not-operated groups received Cognitive Behavioural Therapy (CBT) once a week for eight weeks. CBT is a form of therapy where the aim is to develop person-specific coping strategies [12]. It is the most researched form of psychotherapy and considered to be the current gold standard by many [10]. It has been used to treat pain before [23]. The 37 patients treated with CBT but not operated constitute the control group. Patients were diagnosed and all operations were performed by MD Bo Nyström at the Strängnäs Clinic for Spinal Surgery. To be able to attend the CBT, the patients for this study were recruited from Stockholm area only. The study has been approved by the Stockholm Regional Ethical Committee.

Information on patient expectations was gathered in the form of questionnaires (described in sections 2.1 and 2.2 below). The questionnaires were filled in before randomisation and one year after treatment start. In addition to questions on expectations, the patients filled in following questionnaires both pre- and one year post-treatment; Balanced Inventory for Spinal Disorders (BIS), Oswestry Low Back Disability Index (ODI), 36-Item Short Form Survey In-strument (SF-36) and European Quality of Life Scale (Euro-Qol).

2.1 Pre-treatment - The question setting.

Patients were asked to grade their expectations for four different outcomes (global, back pain, medication, ability to work) all twice. First what they expect the outcome to be if they get operated and separately what they expect the outcome to be if they are randomised to CBT-only control group.

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the cross/line is measured in millimetres. CBT & CBT-only Operated >75% 50-75% 25-50% None >75% 10 7 4 19 50-75% 1 7 4 18 25-50% 0 0 1 3

Table 1: Expected improvement following operation-and-CBT versus CBT-only

Patients were explained that previous studies had not shown fusion operation to be effective for chronic back pain (communication with Bo Nyström), yet most patients believed the oper-ation and CBT together would yield better global outcome than CBT alone (55 patients, 74%, see Table 1). 18 (24%) patients expected a similar outcome from both operation-and-CBT and CBT-only treatments. As both groups received CBT it can be assumed that this second group simply did not believe the operation will help them, which would be reasonable given that pre-vious studies have been unable to prove the operation to be effective. Only one patient (1%) expected the CBT alone to give a better outcome. It is possible that this person did not believe in the efficacy of the operation and was anticipating post-operative complications. Thus we get two groups of patients; those who prefer the operation and those who do not believe it to have an effect. These groups have 55 (74%) and 19 (26%) patients respectively (Table 1). The group of 55 patients positive about the operation also has a subgroup of true believers in the efficacy of the operation, who anticipated a good outcome (>50% improvement) for the operation, but no improvement in the control group. No-one expected their condition to deteriorate after ei-ther of the treatments - which actually happened to ten (14%) of patients - and no-one expected to have no improvement at all if operated. 40 Patients (54%) expected no improvement after being treated with CBT-only. Also, there was a small group of optimists, who believed either treatment would yield a result of over 75% improvement (15% of patients).

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than without it. This reflects the expectations patients had for the global outcome, described above. The curves for highest acceptable level of pain after the two respective treatments (dark red for operated and dark green for control) are close to each other, indicating that highest acceptable pain levels after either of the treatments are similar. It is worth noting that all the curves start from very low level, indicating that there are patients who expecting to be (almost) pain free after which ever treatment. Simultaneously the curves all reach very close to 100, which would suggest that for both treatments there are patients who do not believe their pain will be alleviated.

Figure 1: Cumulative curves for expectations in VAS.

Expectations concerning level of medication was investigated with two identical ordinal scales, in which the answer options were: "No need for pain medication", "Every now and then or low daily dose", "Great need for pain medication". The patients were asked to rate what they expect their need for pain medication to be and what would be the worst acceptable situation for need of pain medication.

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full-time)", "Do not believe I can work again".

2.2 Post-treatment

The patients were asked to evaluate the treatment outcome and how results from the allotted treatment fulfilled their expectations. This was done with two questions for each different out-come (global, back pain, need for medication, ability to work). The first question asked the patients to evaluate their situation (global improvement, pain level (VAS), need of medica-tion, work situation) at the control, one year after treatment. The second asked if the patients felt their expectations had been fulfilled for that particular outcome (Three step ordinal scale: "Yes", "Partly", "No"). In addition the patients were asked to evaluate, if they believed that their pretreatment expectations had an impact on how they judged the treatment outcome. This was done using a three step ordinal scale ("Yes", "Unsure", "No").

Figure 2: Patient reported outcome by treatment group.

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operation can remove the cause of pain for a group of carefully diagnosed patients, plausible. That many in the control group reported deterioration after treatment should not be interpreted as the CBT-treatment causing deterioration, but rather as natural history of the condition (com-munication with Bo Nyström).

This clear global result is also reflected in the pain levels as indicated by the visual analogue scale. This can be seen in Figure 3. It is clear from this Figure that the not operated patients (red dots) have very similar pain levels after and before treatment, being concentrated around the 45 degree line. In contrast, the operated patients (turquoise dots) have similar levels of pain before treatment with the controls, but considerably lower levels of pain after treatment. This is seen from the way values are high before treatment (x-axis), but concentrate under the 25 mm level after treatment (y-axis).

Figure 3: Pain levels pre- and post-treatment measured in VAS.

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Figure 4: Satisfaction with different outcome areas.

One of the last questions patients were asked to answer was "Do you believe that your ex-pectations concerning operation/treatment had an impact on how you evaluated the end result?" ("Yes", "Doubtful", "No"). The results from this question are in Table 2.

Impact? count (%)

Yes 21 (30%)

Doubt 19 (27%)

No 31 (43%)

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3 Methodology - Beta regression and what else?

As was clear from Section 2.2, the operation was highly successful, where as the CBT-alone control treatment was not. In fact, the operation was so effective and results so clear, that there is very little room for analysis. The ordinal primary end variable (Improvement after one year?, see Figure 2) does not allow analysis with for example ordinal regression, as excellent medical outcome causes perfect separation between groups and the analysis would collapse. Dichotomising the outcome would be an option, but it entails undesirable loss of information. So what to do? The medical professional is rejoicing - but the statistician is left with nothing. A possible solution is found when options of analysing post-operative pain levels are researched. The intention is to try beta regression on the (0,1)-bounded visual analogue scale, but it turns out that there have been previous attempts to motivate and use beta regression to ordinal scales as well [33, 40]. Very little is known of the behaviour of beta regression method when continuity of the end variable is violated [40]. However, as the options seem to be that or unsatisfactory dichotomisation, there is desire to give beta regression a go. Thus, the relationship between global treatment outcome and expectations will be investigated using beta regression on the ordinal scale. For this the ordinal scale has to be transformed to (0,1) interval [33]. This has been attempted before, but the properties of such analysis are not well known. Thus they will be investigated.

Beta regression, introduced below, was chosen as a method of interest for the following three reasons. Firstly, as the visual analogue is (0,1) bound, as is the beta distribution, it is a natural choice. Secondly, the beta distribution is very flexible and thus there is a chance to find a good fit. Third, beta distribution is naturally heteroscedastic and can accommodate different patterns of dispersion. Thus not only change in mean, but also change in patterns of dispersion can be analysed, allowing for deeper understanding of the data patterns.

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In some evaluations the scale has been considered superior to other measurement methods [21], but it remains contested [16]. Due to the philosophical aspects of the debate, no conclusions have been, and indeed possibly even can be, reached. Despite these problems, the level of pain post-treatment (VAS) will be used as end point and analysed using beta regression, as hinted above. The performance of this method will then be dissected and compared to other viable methods in a Monte Carlo simulation when there is systematic measurement error in the dependent variable. This is to simulate the possibility that the visual analogue scale does not reflect the underlying pain level correctly.

Beta regression was chosen because it showed promise, but also because options were lim-ited. The dependent variables are bound on an interval. There has been considerable interest in modelling bounded variables in recent years, using for example quantile regression [6] or trans-formation methods [22]. Particularly the beta distribution has proved to be an attractive tool due to its flexibility [33, 40]. There have been several further developments based on it to ac-commodate ever more complicated distributional demands. These include for example the beta rectangular distribution [5], the flexible beta distribution, which is a mixture of two betas [25] and the truncated, inflated beta distribution [31]. Ferrari and Cribari-Neto derived a much used parametrisation of the beta density in the style of the Generalised Linear Model framework, modelling in terms of mean and dispersion, instead of two shape parameters [11]. The pop-ularity of this parametrisation has been aided by the development of a easy-to-use R-package betareg[8].

The PDF of the beta distribution is: y = x

α−1_{(1 − x)}β−1

B(α, β) where: B(α, β) =

Γ(α)Γ(β)

Γ(α + β) (1)

However, to bring this closer to the GLM framework, following parametrization is used:

µ = α

α + β σ =

r 1

α + β + 1 (2)

for µ = (0, 1) and σ = (0, 1). Here the expected value is µ and the variance is σ2µ(1−µ). This parametrization is used by the GAMLSS package in R and thus is also used in this paper.

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might be more homogenous. Beta distribution can naturally accommodate such differences. As a further advantage, the dispersion parameter σ can be modelled as a function of the data. This provides us not only a tool to accommodate heteroskedasticity, but to model differences in dispersion as well.

That beta regression is used on ordinal scales is perhaps surprising. It is quite clear in this case that the assumption of (bounded) continuity is violated. The strategy has been tried before and partly motivated from a philosophical perspective by the boundedness of an ordinal scale that starts at "nothing" and ends with "everything". [40] In essence, it can be argued that assuming there is an underlying latent variable, that would be bounded in nature. The second motivation for use of beta regression is the ability of the beta regression method to model dispersion as well as mean. It can be argued that the changes in dispersion can be just as interesting as changes in mean, and that the nature of ordinal/bounded scales is particularly prone to variations in the dispersion patterns [33]. For example in the data at hand, it can be speculated that the variance of the visual analogue scales and the ordinal scales differ for the different treatment groups (see Figures 2 and 3).

The post-treatment visual analogue scales are zero inflated. That is, they contain zeros. This complicates analysis with beta regression, as the beta distribution is defined on the interval (0,1), not [0,1]. Some methods have been developed to deal with this, but it is an area of ongoing research. The simplest method is to transform the variable from [0,1] to the interval [.5, .95] using the transformation advocated by Smithson and Verkuilen [33]:

y∗ = y(n − 1) + .5

n (3)

where n is the sample size. This is naturally a suboptimal solution, but a simple and practical one. There are also ways to model zero inflated beta distributions, but all these methods assume that the zeros (and/or ones) are generated by a different process than the beta distributed values. [28, 31, 29] That is, they assume that the outcome is either zero, or beta distributed. This cannot be assumed to be the case in our data, as zeros in the visual analogue scales at hand are born out of the same process as the rest of the observations. Zero inflation will be taken into account in the simulation study, where the above transformation is compared to a zero inflated model, which assumes the outcome is either zero or beta distributed. Both the methods have their drawbacks, and neither is assumed to be optimal.

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and dispersion models:

Variables used in mean and dispersion models:

Treatment Operated vs. CBT-only (1/0)

Work BT Working vs. Not working (1/0)

Belief Strong believer in efficacy of operation (0/1) Optimist Expected good results from both treatments (0/1) Pain free Expected to be pain free with both treatments (0/1)

MH BT Emotional well-being (SF-36, score 0-100)

SF BT Social functioning (SF-36, score 0-100)

PF BT Physical functionality (SF-36, score 0-100) Belief*Treatment Believed in operation and operated

The secondary question of interest was if factors can be identified that relate to outcome satisfaction. As discussed above in Introduction (Section 1), evidence from previous studies on surgical outcome, expectations and satisfaction cannot provide a clear hypothesis to work with. Findings have been rather sporadic and sometimes contradictory. Against this backdrop, and considering the rather small sample size, this study will not attempt inference. Rather, the relationship of satisfaction to background information and expectations will be exploratively investigated using the LASSO (Least Absolute Shrinkage and Selection Operator) method. As there are several categorical variables, the group LASSO has to be used [14]. Unfortunately, the method has not been developed yet to handle ordinal dependent variables, and thus the three-step scale ("Satisfied", "Partly", "Not") will be dichotomised to two categories ("Satisfied", "Not completely satisfied").

The LASSO is a shrinkage method which penalises regression coefficient estimates by min-imising a penalised residual sum of squares, using a penalty term λPp

j=1|βj| controlled by the

tuning parameter λ. Here p is the number of free parameters. Depending on the choice of λ, some portion of the smallest coefficients are pushed towards zero, eliminating them completely. Group LASSO is an extension of this method which accommodates categorical variables [14]. It would have been possible to run some other regression method as the amount of variables is not prohibitive. However, when tried, logistic regression did not converge. Also, as the inten-tion is not to do inference, the criteria by which variables are chosen as interesting would be a rather philosophical question. Now we let the LASSO make that choice for us.

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ev-eryone and were unclear. Also, the visual analogue scales with expected/accepted pain levels were dropped and instead pain levels measured in ordinal scales were used. The variables are centred and standardised before group LASSO estimation.

Variables used in group LASSO models:

X1 AT Improvement after one year? (>75%/75-50/25-50/none/deter.) Difference between expected and actual outcome (better/expected/worse) Work BT Working vs. Not working (1/0)

X1 BT Expectation of improvement if operated (5 steps) X2 BT Worst acceptable improvement level if operated (VAS)

X3 BT Expectation of improvement if not operated (CBT-only) (5 steps) X4 BT Worst acceptable improvement level if not operated (VAS) X5 BT Hoped for level of back pain after operation (VAS)

X6 BT Worst acceptable level of back pain after operation (VAS) X7 BT Hoped for level of back pain after CBT-only (VAS) X8 BT Worst acceptable level of back pain after CBT-only (VAS) X9 BT Expected level of medication needed after operation (3 steps) X10 BT Expected level of medication needed after CBT-only (3 steps) X11 BT Worst acceptable level of medication after operation (3 steps) X12 BT Worst acceptable level of medication after CBT-only (3 steps) X17 BT Hoped for level of pain after operation (5 steps)

X18 BT Worst acceptable level of pain after operation (5 steps) X19 BT Hoped for level of pain after CBT-only (5 steps) X20 BT Worst acceptable level of pain after CBT-only (5 steps) VAS BT Level of back pain (VAS)

VAS leg BT Level of leg pain (VAS)

EQ 5T BT European Quality of Life Scale (Index 0-1)

Osw BT Oswestry Low Back Disability Index (Score 0-100%) PF BT Physical functionality (SF-36, score 0-100)

RP BT Role limitations due to physical health (SF-36, score 0-100) BP BT Pain (SF-36, score 0-100)

GH BT General health (SF-36, score 0-100) VT BT Energy/fatigue (SF-36, score 0-100) SF BT Social functioning (SF-36, score 0-100)

RE BT Role limitations due to emotional problems (SF-36, score 0-100) MH BT Emotional well-being (SF-36, score 0-100)

PCS BT Physical component summary (SF-36, score 0-100) MCS BT Mental component summary (SF-36, score 0-100) RoMo BT Roland-Morris Disability Questionnaire (Score 0-24)

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Methods used in analysis:

Endpoint Variable Method Transformation

Outcome Pain (VAS) Beta regression transformation to (0,1)

Global (ordinal) Beta regression transformation to (0,1)

Satisfaction Satisfaction (ordinal) Logistic group LASSO dichotomisation Beliefs Perceived impact (ordinal) Fisher’s exact test dichotomisation

Model selection for the beta regression models is done with backwards elimination starting with the above stated variables. It will be aided by the two-step procedure introduced by, and following recommendations of, Bayer and Cribari-Neto [4].

The data had approximately 2% missing values, which is not a high percentage. However, as the missing values were evenly spread across the observations, they caused larger losses than expected. To keep all observations aboard, the data was singly imputed using appropriate parametric models (logistic, ordinal, predictive mean matching) with the help of R package mice[37]. Single imputation causes overconfidence in results, but due to the small amount of missing values, this should not be a big problem.

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4 Monte Carlo simulations - Are you sure this is OK?

We will use beta regression in two unusual situations; when the outcome is ordinal and when it is a visual analogue scale. The simulations introduced in this section motivate the unusual method choice. To feel confident with the final analysis, majority of simulations are constructed to represent the original data problem in several details.

The nature of visual analogue scale is contested [32, 26]. There have been attempts to investigate the form of the visual analogue scale in experiments [20, 16, 30, 21], but these are always situation specific trials and can never confirm that the scale will function as desired in all situations. This naturally leads to the question; if the scale does not correctly reflect the pain level, how bad can the results get?

For simulations 1.3, 1.4 and 2 the data is generated from beta distribution. In Figures 5 and 6 first the true response variable and then the simulated response variable are plotted. There has been an active attempt to mirror the problem presented by the original data. In the simulated problem, the "control" group has a symmetric distribution, where as the mean of the "treated" group has a mean close to zero. The dispersion is simulated to approximately mimic the results from the post treatment visual analogue scales.

The data generation process is :

f (µ) = β0+ β1x1+ β2x2 (4)

g(σ) = γ0+ γ1x2 (5)

for the mean and dispersion parameters respectively. Where β0 = 0, β1 = 1 and β2 = −1, and

γ0 = 0 and γ1 = 1 . x1 follows a normal distribution, with µ=0 and σ= .1, and x2 is Bernoulli

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Figure 5: True response variable.

Figure 6: Simulated response variable.

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The starting point before creating systematic error is the same for all simulations and was motivated above (see discussion on Figures 5, 6). This is done to guarantee that the results from the simulations will be relevant to the situation at hand.

1. The very first simulation (1.1) is to look at the performance of logistic, ordinal and beta regression when used on ordinal variables simulated using the logistic distribution. Partly following Zou et al. [40] the simulated dependent variable is cut to two categories for the logistic regression and five categories for the beta and ordinal regressions. In this context the logistic and ordinal models are on their home turf, and are naturally expected to do well. The ordinal variable is transformed to zero-one bound using the transformation presented in Section 3 (Methodology) for estimation with beta regression. Logit link is used in all models. In the second part (simulation 1.2), something rather unusual is done; the dependent variable is generated as if from the beta, but actually a zero-one truncated normal distribution (parameter values σ =.3 and µ as defined above) is used. This way it is possible to create a bounded dependent variable, but homoskedastic. In the following scenarios, the dependent variables are generated from the beta distribution, first with constant dispersion (simulation 1.3) and then a dependent dispersion parameter (simulation 1.4). These are analysed with following methods; ordinal, logistic, inflated beta, transformed beta and beta on transformed ordinal scales.

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2.2. Zero avoidance, which causes the scale to jump at small values. This mimics a situation when the anchor is too extreme to be used. [16] See Figure 8 for a visu-alisation. The scenario is created by adding a value to the smallest fourth of the observations. Added values are randomised from uniform distributions with limits [0,.1], [0,.2] and [.1, .3] respectively.

2.3. Zero and one avoidance: centre of scale overrepresented. This scenario is analo-gous to findings by Aubrun [3]. As seen from Figure 9, it is the most extreme, but also the only symmetric transformation. The original data has been multiplied with randomised values from normal distribution truncated at 0 and 2. The distributions are truncated N(1, 0.1), truncated N(1, 0.3) and truncated N(1, 0.5) The observa-tions were first ordered so that the small values are multiplied with values over one, the smaller the original value, the larger the value multiplied with, and the opposite for the other tail. This causes the observations to concentrate towards the centre. As the errors are created with values randomised from the different distributions, they not only distort the values, but also add error to the final result.

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Figure 8: Simulation 2.2

Figure 9: Simulation 2.3

After each desired error has been created, the resulting end variable will be rounded to the nearest two digits. This is done for two reasons; first of all, the visual analogue scale is measured to the accuracy of a millimetre, so the simulation will better reflect the actual mea-surement problem. A second desirable outcome of rounding is the natural creation of zeros in the data. Zero inflation created this way is dependent on the same data generation process as rest of the data points and will thus reflect the creation of zeros in the original setting.

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5 Results

5.1 Comparison of regression methods

Results for simulation 1 are presented in Table 5. The first four columns show result for when ordinal scales are generated from the logistic distribution. As expected, ordinal and logistic methods are clearly more accurate than beta regression methods in terms of bias, hitting very close to the bullseye. For example, for sample size 80 the bias of ordinal and logistic regression is in the second decimal, where as for the beta it is around ten times as large. The methods are better than the beta also in terms of MSE, but the difference is not as great due to the precision of the beta being better, that is, the beta estimates have smaller variance. This is overall a property of the beta regression estimator in comparison to the logistic and ordinal regression methods, and if bias can be reduced, plays clearly to its advantage.

The next four columns ("From truncated normal") contain the results for when the data is generated from zero-one truncated normal. The idea behind this unusual solution was to simulate a truncated dependent variable but completely avoid heteroskedasticity and thus level the playing field for the non-beta models. Indeed, except for the smallest sample size, the ordinal and logistic are better than the beta in terms of bias, and as sample size grows, they catch up with the beta even in terms of MSE, eventually overtaking it when the sample size is 300. By this largest sample size the difference in bias is considerable to the advantage of the ordinal and the logistic.

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wrongly identifying the process causing zero inflation, as the zero inflated beta regression does, is causing problems. Although the results are not reported here, it can also be mentioned that if the amount of zero-inflation is increased, zero-inflated beta regression suffers from increased bias, which the transformed version does not struggle with to the same extent.

However, what is possibly the most interesting detail in this table, is how well the beta regression used on an ordinal variable holds up. It has larger MSE (and mostly larger bias) than the other beta regression methods, but not greatly so. This despite a huge amount of information being lost when the variable is categorised and its continuity heavily violated. In two smallest sample sizes, it is actually the best method in terms of bias. This means that the rather crazy idea of using beta on ordinal variables might pay off when beta distribution can accommodate the dispersion patterns in the data.

The four last columns present results for when the dependent variable is generated from the beta distribution with dispersion defined in Section 3. A similar pattern already seen in the previous columns is repeated here; beta regression methods do clearly better than ordinal or logistic. Using the transformation in equation (3) in combination with beta regression is the best solution again, measured both in bias and MSE, though interestingly enough its bias increases with sample size, which is unexpected.Overall, as sample sizes grow, the bias of beta regression based methods does not decrease. This is also observed when dispersion is constant in the four previous columns. It can be assumed that this is due to the suboptimal ways beta regression is used in this simulation. The zero-inflated method, transforming the variable and most certainly chopping it to pieces and then transforming, all cause bias in themselves.

A clear result of these simulations is that if there is dependency structure between the data and dispersion, the usual ordinal and logistic methods perform clearly worse than any beta re-gression method. Based on these simulations, the following recommendations can be made. - The correct identification of the dependency structure of the variation is essential for the method choice. If there is heteroskedasticity, beta regression takes this into account and per-forms better than logistic or ordinal regression. This holds even when beta regression is used on an ordinal variable.

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Error 2.1.1 Error 2.1.2 Error 2.1.3

n=80 β2 Std.error bias MSE β2 Std.error bias MSE β2 Std.error bias MSE Ordinal -1.5073 0.4490 -0.5073 0.0061 -1.4894 0.4508 -0.4894 0.0059 -1.4938 0.4531 -0.4938 0.0062 Logistic -2.2687 2.2766 -1.2687 0.0288 -2.1733 0.5735 -1.1733 0.0215 -2.0983 0.5602 -1.0983 0.0193 B (inf.) -1.3575 0.2730 -0.3575 0.0027 -1.4216 0.2632 -0.4216 0.0032 -1.4530 0.2590 -0.4530 0.0036 B (transf.) -1.0514 0.2552 -0.0514 0.0009 -1.0833 0.2442 -0.0833 0.0008 -1.0971 0.2390 -0.0971 0.0009 B (ord.) -0.5452 0.2478 0.4548 0.0035 -0.4932 0.2319 0.5068 0.0040 -0.5073 0.2191 0.4927 0.0038

n=80 β2 Std.error bias MSE β2 Std.error bias MSE β2 Std.error bias MSE Ordinal -1.5287 0.4506 -0.5287 0.0068 -1.4839 0.4477 -0.4839 0.0061 -1.1995 0.4347 -0.1995 0.0033 Logistic -2.3823 6.8829 -1.3823 0.0413 -2.4672 25.8382 -1.4672 0.0766 -2.1132 130.8077 -1.1132 0.1646 B (inf.) -0.8446 0.2442 0.1554 0.0012 -0.7107 0.2355 0.2893 0.0018 -0.5130 0.2197 0.4870 0.0037 B (transf.) -0.7890 0.2432 0.2110 0.0014 -0.6604 0.2348 0.3396 0.0023 -0.4624 0.2198 0.5376 0.0044 B (ord.) -0.5603 0.2513 0.4397 0.0035 -0.5419 0.2500 0.4581 0.0037 -0.4417 0.2440 0.5583 0.0049

n=80 β2 Std.error bias MSE β2 Std.error bias MSE β2 Std.error bias MSE Ordinal -1.5334 0.4514 -0.5334 0.0069 -1.5301 0.4508 -0.5301 0.0066 -1.5777 0.4526 -0.5777 0.0070 Logistic -2.4170 7.7207 -1.4170 0.0453 -2.5216 11.9362 -1.5216 0.0570 -2.7123 29.5532 -1.7123 0.0991 B (inf.) -1.0188 0.2479 -0.0188 0.0009 -0.8984 0.2237 0.1016 0.0008 -0.8185 0.2024 0.1815 0.0010 B (transf.) -0.9815 0.2434 0.0185 0.0009 -0.8787 0.2204 0.1213 0.0009 -0.8034 0.1996 0.1966 0.0010 B (ord.) -0.5515 0.2464 0.4485 0.0035 -0.5231 0.2293 0.4769 0.0038 -0.6231 0.1913 0.3769 0.0023

Table 6: Results for the second simulation for n=80.

- Beta regression is more precise than the other methods, which can be a desirable property, depending on the situation.

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clearly the best method and dichotomisation and logistic regression should be avoided if there is indication of throughout systematic bias.

Scenarios 2.2.1 -2.2.3 are reported in the middle rows of Table 13. The error creation process is additive and only concerns the lowest quarter of the simulated values, exaggerating them. Here, as the error scales up and an increasing amount of zeros are pushed within the (0,1) bounds, zero inflated beta regression gains advantage over the other beta regression methods. When no zeros are left, it has become in fact ordinary beta regression, and of beta regression methods it has the smallest bias and MSE. Logistic regression, on the other hand, suffers, as an increasing amount of observations get classified wrongly. It has the biggest bias and MSE throughout. Interestingly, the MSE and bias of ordinal regression actually gets smaller as the error scales up. It seems to be more robust as only small values are affected and most of the observations are correctly classified to their respective groups.

In scenario 2.3 (2.3.1 -2.3.3), reported as the last four rows of Table 13, the values concen-trate to the centre of the scale, with concentration increasing by the sub-scenario. Of the beta regression based methods, here again the zero-inflated model improves its performance as error grows and more and more zeros are pushed to the (0,1) interval. In terms of bias and MSE it is the best method, though the bias increases with the severity of the error. However, the zero inflated beta model and beta regression on transformed variable are very much neck to neck. Logistic regression performs worst without exceptions.

5.2 Medical outcome

As mentioned, 14% of the patients had deteriorated one year after treatment start. Overall, extreme outcomes were overrepresented for the self reported level of change, which is well illustrated in Figure 2. This figure also directly shows how being operated is clearly connected with the outcome.

Based on simulation results presented in Section 5.1, the transformed ordinal variable global outcome is analysed using beta regression. As the analysis of global outcome reveals the dispersion parameter to have a dependency structure with the data, this method is deemed to be the most appropriate for the situation, as discussed in Sections 3 and 5.1. It can also be reminded that ordinal regression would have not worked in this case, due to perfect separation and that using beta instead of logistic regression avoids unnecessary loss of information.

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Estimate Std. Error t value Pr(>|t|) Mean model: (Intercept) 1.361 0.005 248.760 0.000 Treatment -2.722 0.006 -460.666 0.000 Dispersion model: (Intercept) 3.354 0.628 5.344 0.000 Optimist -3.354 0.511 -6.566 0.000 Pain free 2.461 0.369 6.662 0.000 MH BT -0.043 0.008 -5.473 0.000 SF BT 0.102 0.018 5.703 0.000 PF BT -0.149 0.021 -6.971 0.000 Believed 0.524 0.457 1.148 0.255 Treatment -0.617 0.393 -1.571 0.121 Treatment*Believed -3.184 0.712 -4.470 0.000

Table 7: Results from beta regression on Improvement after one year? (ordinal)

It can be read from Table 7, for the mean model only Treatment is statistically significant at 0.05 level in explaining differences in mean. This is perhaps not surprising. For the dispersion model, several of the variables indicating expectations or baseline functionality are significant. (Table 7). This indicates that even though the expectations and baseline functionality of the patients are not affecting changes in the mean, they are related to the patterns of variance. In this parametrisation, increase in dispersion parameter σ value increases the variance, when µ is held constant. Thus, for example, variation is smaller for optimists and those who were operated and believed the operation to yield clearly better results than the control treatment. Better social function (SF BT) at baseline had an inflatory effect on the variance, but not better physical functioning nor emotional well-being (PF BT and MH BT). Expecting to be pain free had also an increasing affect on the variance, which feels contradictory to the results for optimists. However, these two patient groups were distinct, as percentual improvement and actual pain level post-treatment are different things.

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not show great discrepancies, as they should not.

Results for beta regression on the transformed pain level after one year measured with visual analogue scale are to be found in Table 8. It can be clearly seen that the dispersion model when modelling the visual analogue scale is a lot simpler than that for the global outcome in Table 7. This time, no expectation related variables are statistically significant in either of the models. Could it be that attitudes are less related to reporting of pain levels than to that of a more abstract global improvement? Some of the baseline variables are however of interest. The mean model includes treatment assignment and participation in working life pre-treatment, which both indicate lower pain levels after treatment. The dispersion model has picked up physical functioning (PF BT) and emotional well-being (MH BT), with increased PF BT having a decreasing effect on variation in post treatment pain levels and MH BT indicating the opposite direction.

Estimate Std. Error t value Pr(>|t|) Mean model: Intercept 0.860 0.205 4.186 0.000 Treatment -2.030 0.249 -8.163 0.000 Work -0.518 0.235 -2.207 0.031 Dispersion model: Intercept -0.115 0.400 -0.286 0.776 PF BT -0.016 0.006 -2.615 0.011 MH BT 0.012 0.005 2.278 0.026

Table 8: Results from beta regression on Pain level after one year? (VAS)

The residual diagnostic plots for both beta regression models can be found in Appendix B. They indicate that the second model (Table 8), modelling the pain outcome, had better fit. This is not surprising as the continuity of the dependent variable was not violated.

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Satisfaction in different outcome areas

Global result Pain level Medication Working

(Intercept) 1.5652 1.8470 1.1982 1.8967 X1 AT (50-75%) -2.2377 -2.9150 -0.9743 -1.4856 X1 AT (25-50%) -3.1089 -2.4219 -3.0340 -2.1681 X1 AT (None) -3.5045 -3.7252 -3.9575 -1.6267 X1 AT (Deter.) -4.0234 -4.3234 -4.1993 -1.5545 Work 0.2749 0.0000 0.0000 0.0000 Difference (>0) 0.0000 0.0000 0.0000 1.0590 Difference (=0) 0.0000 0.0000 0.0000 0.6301 X1 BT2 0.0000 0.0196 0.0000 0.0000 X1 BT3 0.0000 -0.0304 0.0000 0.0000 X3 BT2 0.0000 0.0000 0.0000 -0.1227 X3 BT3 0.0000 0.0000 0.0000 -0.3213 X3 BT4 0.0000 0.0000 0.0000 -0.1386 X9_BT2 0.0445 0.0000 0.0000 0.0000 X9_BT3 0.0828 0.0000 0.0000 0.0000 X10_BT2 0.0000 0.0000 0.0000 -0.7623 X10_BT3 0.0000 0.0000 0.0000 -0.1710 X11_BT2 0.0000 0.0000 0.6106 0.0000 X11_BT3 0.0000 0.0000 1.5840 0.0000 X17_BT2 0.0483 0.0000 0.0000 0.3135 X17_BT3 1.2100 0.0000 0.0000 -0.0868 X18_BT3 0.0000 0.0555 0.0566 0.0000 X18_BT4 0.0000 1.6281 0.9204 0.0000 X19_BT2 0.0000 0.0000 0.0000 0.2617 X19_BT3 0.0000 0.0000 0.0000 0.2046 X19_BT4 0.0000 0.0000 0.0000 -0.0561 X20_BT2 0.0000 0.0000 0.0000 -0.3582 X20_BT3 0.0000 0.0000 0.0000 -0.9595 X20_BT4 0.0000 0.0000 0.0000 -0.7930 VAS_BT 0.0000 0.0000 0.0000 -0.0154 RP_BT 0.0000 0.0056 0.0012 0.0000 MH_BT 0.0000 0.0000 0.0000 0.0008 PCS_BT 0.0000 0.0066 0.0000 0.0000

Table 9: Results from group LASSO models.

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variable Difference is only in the "work satisfaction" model. It seems that doing better or worse than one expected is not connected to satisfaction. That is surprising and undermines the idea that attainment of expectations leads to satisfaction, proposed by some. [34] Pre-treatment par-ticipation in working life has only been picked up by the global satisfaction model. The same is true of pain level before treatment VAS BT, which only appears in one model. However, sev-eral of the expectations that have been included in the models are pain related: X17 BT to X20 BT all enquire expected or acceptable pain levels after the respective treatments. This is not surprising as pain levels are central to all areas of satisfaction. Variables X9 BT, X10 BT and X12 BT all concern the expected level of medication needed. That three questions from this category pop up in all but the "pain satisfaction" model might indicate that medication could offer a good, concrete way to approach expectations. However, the results for pain expectations and medication are rather sporadic, which makes one doubt if they would hold up if the study were to be repeated. Also, because the ordinal results are treated as categorical, the results are sometimes inconsistent. For example, the coefficients for different groups of variable X19 BT point to different directions. This is very difficult to interpret. Based on these results it can be said that for these patients, only their treatment result is consistently connected to satisfaction in different outcomes. Not even attaining or exceeding their expectations is connected to most areas of satisfaction.

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from the treatment outcome and analysed separately?

Did expectations effect outcome?

Improved Yes Doubt No

>75% 13 6 8

<75% 8 13 23

Fisher’s Exact test:

Alternative hypothesis two sided. p-value: 0.0279

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6 Discussion - Was this a good idea?

Expectations and their implications to treatment outcome in lumbar fusion surgery in the RCT setting have not been looked at before. The clear result from the operation and subsequent division of patients to those who did very well, against those that did not improve at all or even deteriorated, gives us an unique insight to the impact of expectations. This study shows that when the treatment provides clear improvement, it is (almost) only cause affecting the changes in mean of reported treatment outcome. This finding is partly in line with findings by Mannion et al. [2] who found no connection between baseline expectations and post-treatment pain levels.

74% of the patients believed the operation to be more effective than CBT alone, even though they had been informed that previous studies had failed to find the operation effective. A similar finding is done by Toyone et al. [36], as patients rate other reasons as primary for being operated than the one the clinicians have explained is the main medical reason for operation. Mannion et.al. [2] also report overly optimistic patient expectations, but does not find these to have significantly effected post-treatment pain levels. Thus, drawing on these studies it seems that patient expectations for treatment outcomes are sometimes unrealistic and not in line with information and advice given by medical professionals.

The question is, are unrealistic expectations a problem? As mentioned, this study did not find a relationship between the mean outcomes and having high expectations (optimists, pain free, having strong belief in operation and being operated). Mannion et al. reported that "ex-pectations being fulfilled" was the most significant predictor of global outcome [2]. However, correlation not being causality, it can be asked if this might have just indicated that a good outcome is the best predictor of "expectations being fulfilled". If this is the case, we cannot improve satisfaction by moderating expectations, which is sometimes suggested [19]. Findings of this type have also been made by Soroceagu et al. who report that higher fulfilment of ex-pectations led to higher post-operative satisfaction and better outcomes [34]. Mancuso et al. also find that post-operative pain and functionality, among other factors, are connected to fulfil-ment of expectations [24]. Wilson reports similar results [38]. Their study shows that patients seeking surgery to symptoms that are responsive to surgery tend to be more satisfied and have better outcomes. The design of studies on expectations and satisfaction should be improved to minimise the chances of misinterpretations.

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impact the variation patterns of medical outcomes. Several factors were found significant in the dispersion model, with baseline Physical functionality (PF) and Emotional well-being (MH) being included for both post treatment pain levels and global outcome. How these results are to be interpreted in practical terms is a somewhat open question. That better Physical functionality decreases the variation in pain levels might indicate that patients with high functionality at baseline find it easier to relate the outcome to their starting point, for example. However, this information is not necessarily of use in clinical context.

This study also found a connection between improvement levels and patients self-conceived ability to impact their own results. This effect has possibly not been investigated before and should be studied further.

From a statistical point of view, this study highlights the importance of examining dis-persion patterns in bounded dependant variables. It seems that though methods like logistic regression are seen as "safe", it might not perform as well as anticipated when the underlying latent variable is heteroscedastic. Dichotomisation and grouping of values is sometimes seen as a safer way of using unreliable variables, such as the visual analogue scale, but this study ques-tions the validity of that approach and shows that dichotomisation can cause problems other than the sheer loss of information.

The comparisons show that as long as the variance is constant, logistic and ordinal regres-sion are superior in terms of bias in larger sample sizes. Simultaneously, smaller standard error of the beta estimator means that all beta methods tried here consistently have as good or smaller MSEs than logistic or ordinal regression. This quality can be to great advantage, when beta distribution provides a good for the data. In this study, in the situations when the dependent variable was not beta distributed, the beta model still managed to provide results comparable to the ordinal and logistic models in terms of MSE, mostly due to its accuracy. This would indicate that even when beta regression is not the optimal modelling choice, it can still be usable. That beta regression allows the modelling of dispersion patterns gives an added dimension to the analysis. This is not allowed by other methods.

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explosive kind. Interestingly, the type of the error was not crucial for bias induced, and results from the same method were comparable across different error types.

The strengths of this study include: The RCT setting in general, and a situation where treatment has been crucial for the outcome in particular. This provided us with two distinct groups of patients: those whose condition improved radically and whose expectations were met or exceeded and those who did not improve at all. An other strength was that the surgeon has been the same for all patients. This means that possible surgeon induced biases in treatment style and quality, and in information provided to patients, are not a concern.

Limitations are the untested nature of the expectations questionnaires. Others have called for standardised and validated measurement means [1]. Are expectations to become a standard part of treatment assessment, such need to be developed.

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7 Conclusions

This study concludes that expectations were not found to impact treatment mean of outcome, but only variation patterns. Satisfaction was most clearly affected by treatment assignment. The findings of impact of expectations on satisfaction were inconclusive, but indicated that need for medication and general health at baseline should be further researched. Also, level of improvement was connected to the level patients supposed to have impacted their reporting of outcomes.

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Appendix A

Figure 10: Residual diagnostics for final beta regression model on Pain after one year (VAS).

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Appendix B

n=20 β2 Std.error bias MSE β2 Std.error bias MSE β2 Std.error bias MSE

Ordinal -1.9015 4.0685 -0.9015 0.3899 -1.8371 5.1776 -0.8371 0.3288 -2.1149 12.6641 -1.1149 0.5615 Logistic -2.1168 6.4470 -1.1168 0.0447 -2.0490 8.9469 -1.0490 0.0493 -2.0803 1.2623 -1.0803 0.0363 B (inf.) -1.3583 0.5524 -0.3583 0.0292 -1.3887 0.5316 -0.3887 0.0301 -1.4344 0.5242 -0.4344 0.0307 B (transf.) -0.9571 0.4847 0.0429 0.0155 -0.9354 0.4587 0.0646 0.0141 -0.9624 0.4483 0.0376 0.0134 B (ord.) -0.5721 0.4475 0.4279 0.0229 -0.5225 0.4158 0.4775 0.0236 -0.5356 0.3938 0.4644 0.0216 n=80 β2 Std.error bias MSE β2 Std.error bias MSE β2 Std.error bias MSE

Ordinal -1.5073 0.4490 -0.5073 0.0061 -1.4894 0.4508 -0.4894 0.0059 -1.4938 0.4531 -0.4938 0.0062 Logistic -2.2687 2.2766 -1.2687 0.0288 -2.1733 0.5735 -1.1733 0.0215 -2.0983 0.5602 -1.0983 0.0193 B (inf.) -1.3575 0.2730 -0.3575 0.0027 -1.4216 0.2632 -0.4216 0.0032 -1.4530 0.2590 -0.4530 0.0036 B (transf.) -1.0514 0.2552 -0.0514 0.0009 -1.0833 0.2442 -0.0833 0.0008 -1.0971 0.2390 -0.0971 0.0009 B (ord.) -0.5452 0.2478 0.4548 0.0035 -0.4932 0.2319 0.5068 0.0040 -0.5073 0.2191 0.4927 0.0038 n=160 β2 Std.error bias MSE β2 Std.error bias MSE β2 Std.error bias MSE

Ordinal -1.4952 0.3132 -0.4952 0.0023 -1.4560 0.3137 -0.4560 0.0020 -1.4568 0.3150 -0.4568 0.0020 Logistic -2.2044 0.4036 -1.2044 0.0101 -2.0936 0.3890 -1.0936 0.0084 -2.0320 0.3808 -1.0320 0.0077 B (inf.) -1.3671 0.1932 -0.3671 0.0011 -1.4231 0.1858 -0.4231 0.0014 -1.4441 0.1827 -0.4441 0.0015 B (transf.) -1.1035 0.1833 -0.1035 0.0003 -1.1348 0.1754 -0.1348 0.0003 -1.1443 0.1719 -0.1443 0.0003 B (ord.) -0.5242 0.1800 0.4758 0.0017 -0.4659 0.1693 0.5341 0.0020 -0.4830 0.1599 0.5170 0.0019 n=300 β2 Std.error bias MSE β2 Std.error bias MSE β2 Std.error bias MSE

Ordinal -1.4825 0.2272 -0.4825 0.0010 -1.4461 0.2275 -0.4461 0.0009 -1.4425 0.2283 -0.4425 0.0009 Logistic -2.1771 0.2895 -1.1771 0.0049 -2.0662 0.2800 -1.0662 0.0041 -2.0116 0.2744 -1.0116 0.0037 B (inf.) -1.3681 0.1408 -0.3681 0.0005 -1.4282 0.1356 -0.4282 0.0007 -1.4386 0.1333 -0.4386 0.0007 B (transf.) -1.1391 0.1351 -0.1391 0.0001 -1.1776 0.1294 -0.1776 0.0002 -1.1828 0.1269 -0.1828 0.0002 B (ord.) -0.5007 0.1338 0.4993 0.0009 -0.4518 0.1260 0.5482 0.0011 -0.4691 0.1190 0.5309 0.0010

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n=20 β2 Std.error bias MSE β2 Std.error bias MSE β2 Std.error bias MSE Ordinal -1.9644 9.6876 -0.9644 0.3412 -1.9295 4.8789 -0.9295 0.3239 -1.4970 1.6914 -0.4970 0.1972 Logistic -2.2188 138.7253 -1.2188 0.0405 -2.0399 1.2928 -1.0399 0.0318 2.0600 11692.8302 3.0600 18.5744 B (inf.) -0.8738 0.4917 0.1262 0.0174 -0.7787 0.4727 0.2213 0.0177 -0.5719 0.4386 0.4281 0.0228 B (transf.) -0.7967 0.4644 0.2033 0.0164 -0.7060 0.4476 0.2940 0.0176 -0.5216 0.4166 0.4784 0.0236 B (ord.) -0.6040 0.4505 0.3960 0.0219 -0.6098 0.4486 0.3902 0.0212 -0.4999 0.4301 0.5001 0.0258 n=80 β2 Std.error bias MSE β2 Std.error bias MSE β2 Std.error bias MSE Ordinal -1.5287 0.4506 -0.5287 0.0068 -1.4839 0.4477 -0.4839 0.0061 -1.1995 0.4347 -0.1995 0.0033 Logistic -2.3823 6.8829 -1.3823 0.0413 -2.4672 25.8382 -1.4672 0.0766 -2.1132 130.8077 -1.1132 0.1646 B (inf.) -0.8446 0.2442 0.1554 0.0012 -0.7107 0.2355 0.2893 0.0018 -0.5130 0.2197 0.4870 0.0037 B (transf.) -0.7890 0.2432 0.2110 0.0014 -0.6604 0.2348 0.3396 0.0023 -0.4624 0.2198 0.5376 0.0044 B (ord.) -0.5603 0.2513 0.4397 0.0035 -0.5419 0.2500 0.4581 0.0037 -0.4417 0.2440 0.5583 0.0049 n=160 β2 Std.error bias MSE β2 Std.error bias MSE β2 Std.error bias MSE Ordinal -1.4952 0.3127 -0.4952 0.0022 -1.4521 0.3107 -0.4521 0.0020 -1.1745 0.3024 -0.1745 0.0008 Logistic -2.2545 0.4115 -1.2545 0.0109 -2.1640 0.4240 -1.1640 0.0095 -1.3988 4.2057 -0.3988 0.0063 B (inf.) -0.8352 0.1724 0.1648 0.0004 -0.7079 0.1664 0.2921 0.0007 -0.5053 0.1549 0.4947 0.0017 B (transf.) -0.7860 0.1734 0.2140 0.0005 -0.6557 0.1677 0.3443 0.0009 -0.4495 0.1567 0.5505 0.0021 B (ord.) -0.5410 0.1818 0.4590 0.0016 -0.5245 0.1814 0.4755 0.0017 -0.4314 0.1774 0.5686 0.0023 n=300 β2 Std.error bias MSE β2 Std.error bias MSE β2 Std.error bias MSE Ordinal -1.4559 0.2265 -0.4559 0.0009 -1.4578 0.2257 -0.4578 0.0009 -1.1547 0.2191 -0.1547 0.0003 Logistic -2.2061 0.2945 -1.2061 0.0051 -2.1260 0.3007 -1.1260 0.0045 -1.3141 0.3230 -0.3141 0.0007 B (inf.) -0.8195 0.1259 0.1805 0.0002 -0.7086 0.1214 0.2914 0.0003 -0.4967 0.1134 0.5033 0.0009 B (transf.) -0.7695 0.1274 0.2305 0.0002 -0.6555 0.1231 0.3445 0.0004 -0.4331 0.1155 0.5669 0.0011 B (ord.) -0.5033 0.1353 0.4967 0.0009 -0.5130 0.1347 0.4870 0.0009 -0.4106 0.1322 0.5894 0.0012

Table 12: Results for simulations with second error scenario.

n=20 β2 Std.error bias MSE β2 Std.error bias MSE β2 Std.error bias MSE

Ordinal -2.0283 5.5000 -1.0283 0.4144 -2.0141 3.7400 -1.0141 0.3799 -2.1114 6.5631 -1.1114 0.4582 Logistic -2.2093 11.0687 -1.2093 0.0498 -2.2849 9.7384 -1.2849 0.0531 -2.2308 404.8770 -1.2308 0.0494 B (inf.) -1.0553 0.4998 -0.0553 0.0174 -0.9402 0.4553 0.0598 0.0136 -0.8517 0.4093 0.1483 0.0131 B (transf.) -0.9370 0.4645 0.0630 0.0147 -0.8606 0.4276 0.1394 0.0127 -0.7872 0.3846 0.2128 0.0127 B (ord.) -0.5943 0.4370 0.4057 0.0213 -0.5825 0.4145 0.4175 0.0210 -0.5871 0.3676 0.4129 0.0182 n=80 β2 Std.error bias MSE β2 Std.error bias MSE β2 Std.error bias MSE

Ordinal -1.4840 0.2271 -0.4840 0.0010 -1.5009 0.2275 -0.5009 0.0010 -1.5562 0.2283 -0.5562 0.0012 Logistic -2.2498 0.2978 -1.2498 0.0055 -2.3079 0.3048 -1.3079 0.0060 -2.3788 0.3142 -1.3788 0.0067 B (inf.) -0.9982 0.1275 0.0018 0.0001 -0.8974 0.1148 0.1026 0.0001 -0.8204 0.1042 0.1796 0.0001 B (transf.) -0.9877 0.1268 0.0123 0.0001 -0.8921 0.1143 0.1079 0.0001 -0.8164 0.1038 0.1836 0.0002 B (ord.) -0.4998 0.1332 0.5002 0.0009 -0.4803 0.1250 0.5197 0.0010 -0.6543 0.0968 0.3457 0.0004

TREATMENT EXPECTATIONS AND THEIR IMPLICATIONS FOR LUMBAR FUSION SURGERY ON CHRONIC BACK PAIN