Bootstrap Tests for Regression Models

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Discover statistical hypothesis testing, resampling methods, estimation statistics and nonparametric methods in my new book , with 29 step-by-step tutorials and full source code. The bootstrap method is a statistical technique for estimating quantities about a population by averaging estimates from multiple small data samples.

Importantly, samples are constructed by drawing observations from a large data sample one at a time and returning them to the data sample after they have been chosen. This allows a given observation to be included in a given small sample more than once. This approach to sampling is called sampling with replacement. The bootstrap method can be used to estimate a quantity of a population. This is done by repeatedly taking small samples, calculating the statistic, and taking the average of the calculated statistics. We can summarize this procedure as follows:.

The bootstrap is a widely applicable and extremely powerful statistical tool that can be used to quantify the uncertainty associated with a given estimator or statistical learning method. This is done by training the model on the sample and evaluating the skill of the model on those samples not included in the sample.

These samples not included in a given sample are called the out-of-bag samples, or OOB for short. This procedure of using the bootstrap method to estimate the skill of the model can be summarized as follows:. For a given iteration of bootstrap resampling, a model is built on the selected samples and is used to predict the out-of-bag samples. Importantly, any data preparation prior to fitting the model or tuning of the hyperparameter of the model must occur within the for-loop on the data sample.

This is to avoid data leakage where knowledge of the test dataset is used to improve the model. This, in turn, can result in an optimistic estimate of the model skill. A useful feature of the bootstrap method is that the resulting sample of estimations often forms a Gaussian distribution. In additional to summarizing this distribution with a central tendency, measures of variance can be given, such as standard deviation and standard error. Further, a confidence interval can be calculated and used to bound the presented estimate.

This is useful when presenting the estimated skill of a machine learning model. There are two parameters that must be chosen when performing the bootstrap: the size of the sample and the number of repetitions of the procedure to perform. The bootstrap sample is the same size as the original dataset. As a result, some samples will be represented multiple times in the bootstrap sample while others will not be selected at all.

The number of repetitions must be large enough to ensure that meaningful statistics, such as the mean, standard deviation, and standard error can be calculated on the sample. A minimum might be 20 or 30 repetitions. Smaller values can be used will further add variance to the statistics calculated on the sample of estimated values. Ideally, the sample of estimates would be as large as possible given the time resources, with hundreds or thousands of repeats.

We can make the bootstrap procedure concrete with a small worked example. We will work through one iteration of the procedure. We now have our data sample. The example purposefully demonstrates that the same value can appear zero, one or more times in the sample. Here the observation 0. In the case of evaluating a machine learning model, the model is fit on the drawn sample and evaluated on the out-of-bag sample.

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That concludes one repeat of the procedure. It can be repeated 30 or more times to give a sample of calculated statistics. This sample of statistics can then be summarized by calculating a mean, standard deviation, or other summary values to give a final usable estimate of the statistic. We do not have to implement the bootstrap method manually. The scikit-learn library provides an implementation that will create a single bootstrap sample of a dataset. The resample scikit-learn function can be used. It takes as arguments the data array, whether or not to sample with replacement, the size of the sample, and the seed for the pseudorandom number generator used prior to the sampling.

For example, we can create a bootstrap that creates a sample with replacement with 4 observations and uses a value of 1 for the pseudorandom number generator. Unfortunately, the API does not include any mechanism to easily gather the out-of-bag observations that could be used as a test set to evaluate a fit model.

Bootstrapping in general

At least in the univariate case we can gather the out-of-bag observations using a simple Python list comprehension. We can tie all of this together with our small dataset used in the worked example of the prior section. Running the example prints the observations in the bootstrap sample and those observations in the out-of-bag sample. In this tutorial, you discovered the bootstrap resampling method for estimating the skill of machine learning models on unseen data.

Bootstrapping to allow for optimism

Do you have any questions? Ask your questions in the comments below and I will do my best to answer. It provides self-study tutorials on topics like: Hypothesis Tests, Correlation, Nonparametric Stats, Resampling , and much more…. Click to learn more. Michael R. Chernick, Robert A. Yoram Reich, S. Gordon C. Smith, Shaun R. Seaman, Angela M.

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Wood, Patrick Royston, Ian R. Thanks to this post i can finally understand the difference between K-Cross validation and Bootstrap, thanks for the clear explanation. Thanks for the post. I understand what is Bootstrapping machine learning. To me both seem the same. First sample with randomly create a sub-sample from the given data and perform training of model on this. Next, validate the model on left out sample. This sample consisted of women in the Control group and women in the Intervention group.

We will use this data to illustrate methods for simple two group cross-sectional comparisons of HRQoL scores using conventional e.

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The aim of this longitudinal observational study was to evaluate two condition specific and two generic health status questionnaires for measuring HRQoL in patients with Osteoarthritis OA of the Knee, and offer guidance to clinicians and researchers in choosing between them. Patients were recruited from two settings, knee surgery waiting listings and rheumatology clinics. Four self-completion questionnaires including the SF were sent to the subjects on two occasions 6 months apart.

Two hundred and thirty patients returned the questionnaire at initial assessment, consisting of patients awaiting total knee replacement TKR Surgery and patients attending Rheumatology outpatient clinics. At the six-month follow-up assessment, patients returned the questionnaire and in the Surgery and Rheumatology groups respectively. The data used here are based on the patients returning both assessments. We compare the conventional ordinary least squares OLS estimates of standard error SE and Confidence Interval CI for the group regression coefficient with their bootstrap counterparts.

The aim of this RCT, with one year of follow-up, was to establish the relative cost-effectiveness of community leg ulcer clinics that use four layer compression bandaging versus usual care provided by district nurses. Two hundred and thirty-three patients with venous leg ulcers were allocated at random to intervention or control group The intervention consisted of weekly treatment with four layer bandaging in leg ulcer clinic Clinic group or usual care at home by the district nursing service Home group.

The primary outcome was time to complete ulcer healing over the one-year follow-up. We use these data to illustrate the use of summary measures such as the AUC for analysing longitudinal data, using conventional and bootstrap hypothesis tests. The primary efficacy variable in this study was the attainment of American College of Rheumatology ACR criteria for improvement of rheumatoid arthritis.

Secondary efficacy variables included patient assessment of health related quality of life HRQoL. In order to assess the impact of the treatments on patients' health related quality of life, the SF was completed by subjects at seven time-points, Week 0 baseline , Weeks 8, 16, 24, 32, 40, and Week 48 at the end of the study or at the time of premature withdrawal from the trial. Three hundred and six subjects at 48 centres were actually entered into the study.

One hundred and fifty-two subjects receiving methotrexate were randomised to the Neoral treatment group and subjects receiving methotrexate were randomised to the Placebo group. Of the subjects randomised, completed the study. Seventy-nine randomised subjects discontinued from the study prior to completion.

We use these data to illustrate more complex statistical models for analysing longitudinal data e.

Figures 1 and 2 show the histograms of the SF dimension scores at six weeks post-natally for Intervention and Control groups. The graphs clearly show the bounded, skewed and discrete nature of the data for the SF from this study. Table 1 shows the two sample t -test with equal variances and Mann-Whitney MW comparisons of the eight SF dimension scores. The only major contrast between the interpretation of the results of the MW and t -tests is on the BP and PF dimensions, where the former test suggests a difference and later not.

The last column of Table 1 also shows the results of a bootstrap hypothesis test for comparing two means. It compares and contrasts the results of the p -values from a bootstrap hypothesis tests with the p -values from the standard two sample t -test with equal variances, and the MW test. So in this example dataset there appears to be little advantage in using the bootstrap hypothesis tests compared to conventional hypothesis tests, such as the t- test, for testing equality of means. A major limitation of non-parametric methods, such as the MW test, is that they do not allow for the estimation of confidence intervals for parameters or allow for the adjustment of confounding variables such as baseline covariates.

One way to estimate non-parametric CIs is via the bootstrap method. The estimates and lengths of the CIs are almost identical. Table 2 also shows that the shape of the BC a CIs is almost symmetric about the point estimate of the mean difference except for the RE dimension, where there is some evidence of asymmetry. So again in this example dataset there appears little advantage in using the bootstrap BC a confidence intervals compared to conventional methods of confidence interval estimation.

The bootstrap and Normal confidence intervals are calculated for a characteristic of the distributions for example mean difference. The groups may have differences in distributions but similar characteristics e. When a hypothesis is tested using the bootstrap, the resampling is carried out assuming the null hypothesis H 0 is true. Whereas when confidence intervals for mean differences between two groups are estimated the resampling is carried out separately for each group.

A useful analogy is with the comparison of proportions in two independent groups.

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Here the standard error for the hypothesis test is different to the standard error of the difference between the observed proportions used for estimating a confidence interval [Page 45, [ 19 ]]. Table 3 shows the baseline socio-demographic and HRQoL characteristics of the two groups of OA patients those awaiting total knee replacement surgery Surgical and those having pharmacological treatment Rheumatology.

The group of patients awaiting surgery is significantly older and has significantly more men than the Rheumatology group. The Surgical group has significantly lower levels of PF prior to total knee replacement surgery than the Rheumatology group. A positive value for the regression coefficient indicates the Surgery group has a better mean HRQoL at six months follow-up than the Clinic group after adjustment for the other covariates.

Table 4 compares the OLS and bootstrap standard errors and confidence interval estimates for the group coefficient from the OA Knee data. All models include age, baseline HRQoL and gender as covariates in the regression. For the bootstrap methods the standard errors are the standard deviations of the coefficients from the bootstrap re-samples.

For ease of interpretation and comparison only the estimates for the group coefficient are shown. As can be seen from Table 4 the standard error estimates are almost identical for the three methods. Similarly the length of the confidence intervals is virtually the same for all three methods. Although the bootstrap CIs tend to be asymmetric about the point-estimate of the regression coefficient. Qualitatively all of the intervals from the three methods either include or exclude zero so the interpretation of the group regression coefficient is the same.

Therefore, again in this example dataset, there appears to be little advantage in using bootstrap case or model based re-sampling to estimate standard errors and confidence intervals compared to conventional methods of confidence interval estimation from the OLS multiple regression model. If we set the time units for the AUC calculation as a fraction of a year, then an AUC value of implies the leg ulcer patient has been in "good health" for the entire month follow-up period. Conversely an AUC value of 0 implies the leg ulcer patient has been in "poor health" for the entire month follow-up period.

Table 7 [See additional file 1 ] gives the results of simple comparisons of differences in mean AUC between the groups using the two independent samples t -test, the MW test and the bootstrap hypothesis test. The p -values from the t -test and the ASL from the bootstrap hypothesis tests are very similar. None of the p -values for the eight SF dimensions are less than 0. Therefore there is no reliable statistical evidence to suggest a difference in mean AUC between the Clinic and Home treated leg-ulcer patients. The table also contrasts the Normal theory based CI estimates from the t -test with the bootstrap BC a limits.

The lengths of the intervals are very similar, although the bootstrap BC a intervals tend to have a non-symmetric shape. The modelling of longitudinal data takes into account the fact that successive HRQoL assessments by a particular subject are likely to be correlated. Marginal models are appropriate when inferences about the population average are the focus. For example, in a clinical trial the average difference between control and treatment is most important, not the difference for any one individual.

In a marginal model, the regression of the response on explanatory variables is modelled separately from the within-person correlation.

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The marginal model is an extension of the linear regression model used with the OA Knee data. Longitudinal models require the specification of the auto- or serial correlation , which is the strength of the association between successive longitudinal measurements of a single HRQoL variable on the same patient. Several underlying patterns of the auto-correlation matrix are used in the modelling of HRQoL data. The error structure is independent sometimes termed random if the off diagonal terms of the auto-correlation matrix are zero. The repeated HRQoL observations on the same subject are then independent of each other, and can be regarded as though they were observations from different individuals.

On the other hand, if all the correlations are approximately equal or uniform then the matrix of correlation coefficients is termed exchangeable , or compound symmetric. This means that we can re-order exchange the successive observations in any way we choose in our data file without affecting the pattern in the correlation matrix.

As the time or lag between successive observations increases, the auto-correlation between the observations decreases. A correlation matrix of this form is said to have an autoregressive structure sometimes called multiplicative or time series. Table 5 summarises the resulting 21 auto-correlation pairs for the assessments until week The pattern of the observed auto-correlation matrix, gives a guide to the so-called error structure associated with the successive HRQoL measurements. Table 5 shows that the autocorrelation coefficients range between 0. The pattern of values suggests that the assumption of compound symmetry is not unreasonable.

The process of fitting marginal models using GEE begins by assuming the simple independence form for the autocorrelation matrix, and fitting the model as if each assessment were from a different patient. Once this model is obtained the corresponding residuals are calculated and these are then used to estimate the autocorrelation matrix assuming it is of the exchangeable or autoregressive type. This matrix is then used to fit the model again, the residuals are once more calculated, and the autocorrelation matrix obtained.

The iteration process is repeated until the corresponding regression coefficients that are obtained in the successive models converge or differ little on successive occasions [ 1 ]. Fayers and Machin [Pages —, [ 1 ]] and Diggle et al [ 10 ] emphasise the importance of graphical presentation of longitudinal data prior to modelling. The curves for some dimensions of the SF overlap e. For other dimensions such as BP, V and SF there is some evidence to suggest that for later HRQoL measurements the curves are parallel and that the mean difference between treatments is now fairly constant.

It is therefore important to test for any such interaction in any regression model. Fortunately, with the marginal model approach this is relatively easy to do and simply involves the addition of an extra regression coefficient to the model. The marginal regression models were fitted in STATA [ 12 ] using the xtgee command with an identity link function link iden and the robust standard errors option.

The observed correlation matrices in Table 5 clearly show the off-diagonal terms are non-zero and that the assumption of an independent auto-correlation matrix for the marginal model is unrealistic. We will not consider models with an independent auto-correlation structure and will concentrate on reporting the results of models with an exchangeable correlation.

None of the interaction term coefficients for the eight SF dimensions were statistically significant from zero. Therefore we will only report the results of the simpler model 2 , without the interaction term. The beauty of the marginal model and the GEE methodology is that it is very flexible and can in principle deal with all the observed data from a HRQoL study. The subjects are not required to have exactly the same numbers of assessments, and even the assessments can be made at variable times. The latter allows the modelling to proceed even if a subject misses a HRQoL assessment.

So it seems unrealistic and unreasonable to use bootstrap resampling methods for marginal models that can only utilise a balanced data set, with equally spaced QoL assessments. Since we are interested in fitting a marginal model and we are likely to have an unbalanced dataset with unequal observations per subject we used simple bootstrap case-resampling.

Figure 4 shows the estimated within subject correlation matrices for the eight dimensions of the SF if we fit the longitudinal model and assume a compound symmetric structure. The lower diagonal gives the observed matrix before the model fitting. The fitted autocorrelations ranged from 0. On the whole, the model correlation estimates tend to be lower than the actual observed autocorrelations, for HRQoL assessments that are close together.

Conversely the model correlation estimates tend to be larger than the observed correlations for HRQoL observations further apart in time. It will usually be the case that after model fitting the autocorrelations will appear to have been reduced [ 1 ].

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The observed deviations between the fitted model and observed autocorrelations are not too great, suggesting that the assumption of compound symmetry is not unreasonable Figure 4. Busqueda avanzada. Compartir en: Facebook Twitter. Bootstrap tests for regression models Autor: Godfrey, Leslie. Resumen Modern computer systems are now so powerful that they can be used to carry out simulation-based statistical investigations without involving delays or the need to access high levels of equipment.