# Calculating the sample size required for developing a clinical prediction model

BMJ 2020; 368 doi: https://doi.org/10.1136/bmj.m441 (Published 18 March 2020) Cite this as: BMJ 2020;368:m441- Richard D Riley, professor of biostatistics1,
- Joie Ensor, lecturer in biostatistics1,
- Kym I E Snell, lecturer in biostatistics1,
- Frank E Harrell Jr, professor of biostatistics2,
- Glen P Martin, lecturer in health data sciences3,
- Johannes B Reitsma, associate professor4,
- Karel G M Moons, professor of clinical epidemiology4,
- Gary Collins, professor of medical statistics5,
- Maarten van Smeden, assistant professor4 5 6

^{1}Centre for Prognosis Research, School of Primary, Community and Social Care, Keele University, Staffordshire ST5 5BG, UK^{2}Department of Biostatistics, Vanderbilt University School of Medicine, Nashville TN, USA^{3}Division of Informatics, Imaging and Data Science, Faculty of Biology, Medicine and Health, University of Manchester, Manchester Academic Health Science Centre, Manchester, UK^{4}Julius Center for Health Sciences, University Medical Center Utrecht, Utrecht, Netherlands^{5}Centre for Statistics in Medicine, Nuffield Department of Orthopaedics, Rheumatology and Musculoskeletal Sciences, University of Oxford, Oxford, UK^{6}Department of Clinical Epidemiology, Leiden University Medical Center Leiden, Netherlands

- Correspondence to: R D Riley r.riley{at}keele.ac.uk (or @richard_d_riley on Twitter)

- Accepted 19 December 2019

Clinical prediction models aim to predict outcomes in individuals, to inform diagnosis or prognosis in healthcare. Hundreds of prediction models are published in the medical literature each year, yet many are developed using a dataset that is too small for the total number of participants or outcome events. This leads to inaccurate predictions and consequently incorrect healthcare decisions for some individuals. In this article, the authors provide guidance on how to calculate the sample size required to develop a clinical prediction model.

#### Summary points

Patients and healthcare professionals require clinical prediction models to accurately guide healthcare decisions

Larger sample sizes lead to the development of more robust models

Data should be of sufficient quality and representative of the target population and settings of application

It is better to use all available data for model development (ie, avoid data splitting), with resampling methods (such as bootstrapping) used for internal validation

When developing prediction models for binary or time-to-event outcomes, a well known rule of thumb for the required sample size is to ensure at least 10 events for each predictor parameter

The actual required sample size is, however, context specific and depends not only on the number of events relative to the number of candidate predictor parameters but also on the total number of participants, the outcome proportion (incidence) in the study population, and the expected predictive performance of the model

We propose to use such information to tailor sample size requirements to the specific setting of interest, with the aim of minimising the potential for model overfitting while targeting precise estimates of key parameters

Our proposal can be implemented in a four step procedure and is applicable for continuous, binary, or time-to-event outcomes

The pmsampsize package in Stata or R allows researchers to implement the procedure

Clinical prediction models are needed to inform diagnosis and prognosis in healthcare.123 Well known examples include the Wells score,45 QRISK,67 and the Nottingham prognostic index.89 Such models allow health professionals to predict an individual’s outcome value, or to predict an individual’s risk of an outcome being present (diagnostic prediction model) or developed in the future (prognostic prediction model). Most prediction models are developed using a regression model, such as linear regression for continuous outcomes (eg, pain score), logistic regression for binary outcomes (eg, presence or absence of pre-eclampsia), or proportional hazards regression models for time-to-event data (eg, recurrence of venous thromboembolism).10 An equation is then produced that can be used to predict an individual’s outcome value or outcome risk conditional on his or her values of multiple predictors, which might include basic characteristics such as age, weight, family history, and comorbidities; biological measurements such as blood pressure and biomarkers; and imaging or other test results. Supplementary material S1 shows examples of regression equations.

Developing a prediction model requires a development dataset, which contains data from a sample of individuals from the target population, containing their observed predictor values (available at the intended moment of prediction11) and observed outcome. The sample size of the development dataset must be large enough to develop a prediction model equation that is reliable when applied to new individuals in the target population. What constitutes an adequately large sample size for model development is, however, unclear,12 with various blanket “rules of thumb” proposed and debated.1314151617 This has created confusion about how to perform sample size calculations for studies aiming to develop a prediction model.

In this article we provide practical guidance for calculating the sample size required for the development of clinical prediction models, which builds on our recent methodology papers.1314151618 We suggest that current minimum sample size rules of thumb are too simplistic and outline a more scientific approach that tailors sample size requirements to the specific setting of interest. We illustrate our proposal for continuous, binary, and time-to-event outcomes and conclude with some extensions.

## Moving beyond the 10 events per variable rule of thumb

In a development dataset, the effective sample size for a continuous outcome is determined by the total number of study participants. For binary outcomes, the effective sample size is often considered about equal to the minimum of the number of events (those with the outcome) and non-events (those without the outcome); time-to-event outcomes are often considered roughly equal to the total number of events.10 When developing prediction models for binary or time-to-event outcomes, an established rule of thumb for the required sample size is to ensure at least 10 events for each predictor parameter (ie, each β term in the regression equation) being considered for inclusion in the prediction model equation.192021 This is widely referred to as needing at least 10 events per variable (10 EPV). The word “variable” is, however, misleading as some predictors actually require multiple β terms in the model equation—for example, two β terms are needed for a categorical predictor with three categories (eg, tumour grades I, II, and III), and two or more β terms are needed to model any non-linear effects of a continuous predictor, such as age or blood pressure. The inclusion of interactions between two or more predictors also increases the number of model parameters. Hence, as prediction models usually have more parameters than actual predictors, it is preferable to refer to events per candidate predictor parameter (EPP). The word candidate is important, as the amount of model overfitting is dictated by the total number of predictor parameters considered, not just those included in the final model equation.

The rule of at least 10 EPP has been widely advocated perhaps as a result of its simplicity, and it is regularly used to justify sample sizes within published articles, grant applications, and protocols for new model development studies, including by ourselves previously. The most prominent work advocating the rule came from simulation studies conducted in the 1990s,192021 although this work actually focused more on the bias and precision of predictor effect estimates than on the accuracy of risk predictions from a developed model. The adequacy of the 10 EPP rule has often been debated. Although the rule provides a useful starting point, counter suggestions include either lowering the EPP to below 10 or increasing it to 15, 20, or even 50.102223242526 These inconsistent recommendations reflect that the required EPP is actually context specific and depends not only on the number of events relative to the number of candidate predictor parameters but also on the total number of participants, the outcome proportion (incidence) in the study population, and the expected predictive performance of the model.1314151617 This finding is unsurprising as sample size considerations for other study designs, such as randomised trials of interventions, are all context dependent and tailored to the setting and research question. Rules of thumb have also been advocated in the continuous outcome setting, such as two participants per predictor,27 but these share the same concerns as for 10 EPP.16

## Sample size calculation to ensure precise predictions and minimise overfitting

Recent work by van Smeden et al1314 and Riley et al1516 describe how to calculate the required sample size for prediction model development, conditional on the user specifying the overall outcome risk or mean outcome value in the target population, the number of candidate predictor parameters, and the anticipated model performance in terms of overall model fit (R^{2}). These authors’ approaches can be implemented in a four step procedure. Each step leads to a sample size calculation, and ultimately the largest sample size identified is the one required. We describe these four steps, and, to aid general readers, provide the more technical details of each step in the figures.

### Step 1: What sample size will produce a precise estimate of the overall outcome risk or mean outcome value?

Fundamentally, the sample size must allow the prediction model’s intercept to be precisely estimated, to ensure that the developed model can accurately predict the mean outcome value (for continuous outcomes) or overall outcome proportion (for binary or time-to-event outcomes). A simple way to do this is to calculate the sample size needed to precisely estimate (within a small margin of error) the intercept in a model when no predictors are included (the null model).15Figure 1 shows the calculation for binary and time-to-event outcomes, and we generally recommend aiming for a margin of error of ≤0.05 in the overall outcome proportion estimate. For example, with a binary outcome that occurs in half of individuals, a sample size of at least 385 people is needed to target a confidence interval of 0.45 to 0.55 for the overall outcome proportion, and thus an error of at most 0.05 around the true value of 0.5. To achieve the same margin of error with outcome proportions of 0.1 and 0.2, at least 139 and 246 participants, respectively, are required.

For time-to-event outcomes, a key time point needs to be identified, along with the anticipated outcome event rate. For example, with an anticipated event rate of 10 per 100 person years of the entire follow-up, the sample size must include a total of 2366 person years of follow-up to ensure an expected margin of error of ≤0.05 in the estimate of a 10 year outcome probability of 0.63, such that the expected confidence interval is 0.58 to 0.68.

For continuous outcomes, the anticipated mean and variance of outcome values must be prespecified, alongside the anticipated percentage of variation explained by the prediction model (see supplementary material S2 for details).16

### Step 2: What sample size will produce predicted values that have a small mean error across all individuals?

In addition to predicting the average outcome value precisely (see step 1), the sample size for model development should also aim for precise predictions across the spectrum of predicted values. For binary outcomes, van Smeden et al use simulation across a wide range of scenarios to evaluate how the error of predicted outcome probabilities from a developed model depends on various characteristics of the development dataset sampled from a target population.14 They found that the number of candidate predictor parameters, total sample size, and outcome proportion were the three main drivers of a model’s mean predictive accuracy. This led to a sample size formula (fig 2) to help ensure that new prediction models will, on average, have a small prediction error in the estimated outcome probabilities in the target population (as measured by the mean absolute prediction error, MAPE). The calculation requires the number of candidate predictor parameters and the anticipated outcome proportion in the target population to be prespecified. For example, with 10 candidate predictor parameters and an outcome proportion of 0.3, a sample size of at least 461 participants and 13.8 EPP is required to target a mean absolute error of 0.05 between observed and true outcome probabilities (see fig 2 for calculation). The calculation is available as an interactive tool (https://mvansmeden.shinyapps.io/BeyondEPV/) and applicable to situations with 30 or fewer candidate predictors. Ongoing work aims to extend to larger numbers of candidate predictors and also to time-to-event outcomes.

For continuous outcomes, accurate predictions across the spectrum of predicted values require the standard deviation of the residuals to be precisely estimated.1016 Supplementary material S3 shows that to target a less than 10% multiplicative error in the estimated residual standard deviation, the required sample size is simply 234+P, where P is the number of predictor parameters considered.

### Step 3: What sample size will produce a small required shrinkage of predictor effects?

Our third recommended step is to identify the sample size required to minimise the problem of overfitting.28 Overfitting is when a developed model’s predictions are more extreme than they ought to be for individuals in a new dataset from the same target population. For example, an overfitted prediction model for a binary outcome will give a predicted outcome probability too close to 1 for individuals with a higher than the average outcome probability and too close to 0 for individuals with a lower than the average outcome probability. Overfitting notably occurs when the sample size is too small. In particular, when the number of candidate predictor parameters is large relative to the number of participants in total (for continuous outcomes) or to the number of participants with the outcome event (for binary or time-to-event outcomes). A consequence of overfitting is that a developed model’s apparent predictive performance (as observed in the development dataset itself) will be optimistic (ie, too high), and its actual predictive performance in new data from the same target population will be lower (ie, worse).

Shrinkage (also known as penalisation or regularisation) methods deal with the problem of overfitting by reducing the variability in the developed model’s predictions such that extreme predictions (eg, predicted probabilities close to 0 or 1) are pulled back toward the overall average.293031323334 However, there is no guarantee that shrinkage will fully overcome the problem of overfitting when developing a prediction model. This is because the shrinkage or penalty factors (which dictate the magnitude of shrinkage required) are also estimated from the development dataset and, especially when the sample size is small, are often imprecise and so fail to tackle the magnitude of overfitting correctly in a particular application.30 Furthermore, a negative correlation tends to occur between the estimated shrinkage required and the apparent performance of a model. If the apparent model performance is excellent simply by chance, the required shrinkage is typically estimated too low.30 Thus, ironically, in those situations when overfitting is of most concern (and thus shrinkage is most urgently needed), the prediction model developer has insufficient assurance in selecting the proper amount of shrinkage to cancel the impact of overfitting.

Riley et al therefore suggest identifying the sample size and number of candidate predictors that correspond to a small amount of desired shrinkage (≤10%) during model development.1516 The sample size calculation (fig 3) requires the researcher to prespecify the number of candidate predictor parameters and, for binary or time-to-event outcomes, the anticipated outcome proportion or rate, respectively, in the target population. In addition, a (conservative) value for the anticipated model performance is required, as defined by the Cox-Snell R squared statistic (R^{2}_{cs}).1535 The anticipated value of R^{2}_{cs} is important because it reflects the signal:noise ratio, which has an impact on the estimation of multiple parameters and the potential for overfitting. When the signal:noise ratio is anticipated to be high (eg, R^{2}_{cs} is close to 1 for a prediction model with a continuous outcome), true patterns are easier to detect and so overfitting is less of a concern, such that more predictor parameters can be estimated. However, when the signal:noise ratio is low (ie, R^{2}_{cs} is anticipated to be close to 0), true patterns are harder to identify and there is more potential for overfitting, such that fewer predictor parameters can be estimated reliably.

In the continuous outcome setting, R^{2}_{cs} is simply the coefficient of determination R^{2}, which quantifies the proportion of the variance of outcome values that is explained by the prediction model and thus is between 0 and 1. For example, when developing a prediction model for a continuous outcome with up to 30 predictor parameters and an anticipated R^{2}_{cs} of 0.7, a sample size of 206 participants is required to ensure the expected shrinkage is 10% (see supplementary material S4 for full calculation). This corresponds to about seven participants for each predictor parameter considered.

The R^{2}_{cs} statistic generalises to non-continuous outcomes and allows sample size calculations to minimise the expected shrinkage when developing a prediction model for binary and time-to-event outcomes (fig 3). For example, when developing a new logistic regression model with up to 20 candidate predictor parameters and an anticipated R^{2}_{cs} of at least 0.1, a sample size of 1698 participants is required to ensure the expected shrinkage is 10% (see fig 3 for full calculation). If the target setting has an outcome proportion of 0.3, this corresponds to an EPP of 25.5. The required sample size and EPP are sensitive to the choice of R^{2}_{cs}, with lower anticipated values of R^{2}_{cs} leading to higher required sample sizes. Therefore, a conservative choice of R^{2}_{cs} is recommended (fig 4).

As in sample size calculations for randomised trials evaluating intervention effects, external evidence and expert opinion are required to inform the values that need specifying in the sample size calculator. Figure 4 provides guidance for specifying R^{2}_{cs}. Importantly, unlike for continuous outcomes when R^{2}_{cs} is bounded between 0 and 1, the R^{2}_{cs} is bounded between 0 and max(R^{2}_{cs}) for binary and time-to-event outcomes. The max(R^{2}_{cs}) denotes the maximum possible value of R^{2}_{cs}, which is dictated by the overall outcome proportion or rate in the development dataset and is often much less than 1. Supplementary material S5 shows the calculation of max(R^{2}_{cs}). For logistic regression models with outcome proportions of 0.5, 0.4, 0.3, 0.2, 0.1, 0.05, and 0.01, the corresponding max(R^{2}_{cs}) values are 0.75, 0.74, 0.71, 0.63, 0.48, 0.33, and 0.11, respectively. Thus the anticipated R^{2}_{cs} might be small, even for a model with potentially good performance.

### Step 4: What sample size will produce a small optimism in apparent model fit?

The sample size should also ensure a small difference in the developed models apparent and optimism adjusted values of R^{2}_{Nagelkerke} (ie, R^{2}_{cs}/max(R^{2}_{cs})), as this is a fundamental overall measure of model fit.1038 The apparent R^{2}_{Nagelkerke} value is simply the model’s observed performance in the same data as used to develop the model, whereas the optimism adjusted R^{2}_{Nagelkerke} value is a more realistic (approximately unbiased) estimate of the model’s fit in the target population. The sample size calculations are shown in supplementary material S6 for continuous outcomes and in figure 5 for binary and time-to-event outcomes. As before, they require the user to specify the anticipated R^{2}_{cs} and the max(R^{2}_{cs}), as described in figure 4. For example, when developing a logistic regression model with an anticipated R^{2}_{cs} of 0.2, and in a setting with an outcome proportion of 0.05 (such that the max(R^{2}_{cs}) is 0.33), 1079 participants are required to ensure the expected optimism in the apparent R^{2}_{Nagelkerke} is just 0.05 (see figure 5 for calculation).

## Recommendations and software

Box 1 summarises our recommended steps for calculating the minimum sample size required for prediction model development. This involves four calculations for binary outcomes (B1 to B4), three for time-to-event outcomes (T1 to T3), and four for continuous outcomes (C1 to C4). To implement the calculations, we have written the pmsampsize package for Stata and R. The software calculates the sample size needed to meet all the criteria listed in box 1 (except B2, which is available at https://mvansmeden.shinyapps.io/BeyondEPV/), conditional on the user inputting values of required parameters such as the number of candidate predictors, the anticipated outcome proportion in the target population, and the anticipated R^{2}_{cs}. The calculations are especially helpful when prospective data collection (eg, new cohort study) are required before model development; however, they are also relevant when existing data are available to guide the number of predictors that can be considered.

#### Box 1 Recommendations for calculating the sample size needed when developing a clinical prediction model for continuous, binary, and time-to-event outcomes

To increase the potential for developing a robust prediction model, the sample size should be at least large enough to minimise model overfitting and to target sufficiently precise model predictions

#### Binary outcomes

*For binary outcomes, ensure the sample size is enough to:*Estimate the overall outcome proportion with sufficient precision (use equation in figure 1) (B1)

Target a small mean absolute prediction error (use equation in figure 2, if number of predictor parameters is ≤30) (B2)

Target a shrinkage factor of 0.9 (use equation in figure 3) (B3)

Target small optimism of 0.05 in the apparent R

^{2}_{Nagelkerke}(use equation in figure 5) (B4)

#### Time-to-event outcomes

*For time-to-event outcomes, ensure the sample size is enough to:*Estimate the overall outcome proportion with sufficient precision at one or more key time-points in follow-up (use equation in figure 1) (T1)

Target a shrinkage factor of 0.9 (use equation in figure 3) (T2)

Target small optimism of 0.05 in the apparent R

^{2}_{Nagelkerke}(use equation in figure 5) (T3)

#### Continuous outcomes

*For continuous outcomes, ensure the sample size is enough to:*Estimate the model intercept precisely (see supplementary material 1) (C1)

Estimate the model residual variance with sufficient precision (see supplementary material 2) (C2)

Target a shrinkage factor of 0.9 (use equation in figure 3) (C3)

Target small optimism of 0.05 in the apparent R

^{2}_{Nagelkerke}(use equation in figure 5) (C4)

These approaches require researchers to specify the anticipated overall outcome risk or mean outcome value in the target population, the number of candidate predictor parameters, and the anticipated model performance in terms of overall model fit (R

^{2}_{cs}). When the choice of values is uncertain, we generally recommend being conservative and so taking those values (eg, smallest R^{2}_{cs}) that give larger sample sizesWhen an existing dataset is already available (such that sample size is already defined), the calculations can be used to identify if the sample size is sufficient to estimate the overall outcome risk or the mean outcome value, and how many predictor parameters can be considered before overfitting becomes a concern

## Applied examples

We now illustrate the recommendations in box 1 by using three examples.

### Example 1: Binary outcome

North et al developed a model predicting pre-eclampsia in pregnant women based on clinical predictors measured at 15 weeks’ gestation,43 including vaginal bleeding, age, previous miscarriage, family history, smoking, and alcohol consumption. The model included 13 predictor parameters and had a C statistic of 0.71. Emerging research aims to improve this and other pre-eclampsia prediction models by including additional predictors (eg, biomarkers and ultrasound measurements).

As the outcome is binary, the sample size calculation for a new prediction model needs to examine criteria B1 to B4 in box 1. This requires us to input the overall proportion of women who will develop pre-eclampsia (0.05) and the number of candidate predictor parameters (assumed to be 30 for illustration). For an outcome proportion of 0.05, the max(R^{2}_{cs}) value is 0.33 (see supplementary material S5). If we assume, conservatively, that the new model will explain 15% of the variability, the anticipated R^{2}_{cs} value is 0.15×0.33=0.05. Now we can check criteria B1, B3, and B4 by typing in Stata:

This indicates that at least 5249 women are required, corresponding to 263 events and an EPP of 8.75. This is driven by criterion B3, to ensure the expected shrinkage required is just 10% (to minimise the potential overfitting). To check criterion B2 in box 1, we can apply the formula in figure 2. This suggests that 544 women are needed to target a mean absolute error in predicted probabilities of ≤0.05. This is much lower than the 5249 women needed to meet criterion B3.

If recruiting 5249 women is impractical (eg, because of time, cost, or practical constraints for data collection), the sample size required can be reduced by identifying a smaller number of candidate predictors (eg, based on existing evidence from systematic reviews44). For example, with 20 rather than 30 candidate predictors, the required sample size to meet all four criteria is at least 3500 women and 175 events (still 8.75 EPP).

### Example 2: Time-to-event outcome

Many prognostic models are available for the risk of a recurrent venous thromboembolism (VTE) after cessation of treatment for a first VTE.45 For example, the model of Ensor et al included predictors of age, sex, site of first clot, D-dimer level, and the lag time from cessation of treatment until measurement of D-dimer (often around 30 days).46 The model’s C statistic was 0.69 and the adjusted R^{2}_{cs} was 0.051 (corresponding to 8% of the total variation). Emerging research aims to extend such models by including additional predictors.

The sample size required for a new model must at least meet criteria T1 to T3.15 This requires us to input a key time point for prediction of VTE recurrence risk (eg, two years), alongside the number of candidate predictor parameters (n=30), the anticipated mean follow-up (2.07 years), and outcome event rate (0.065, or 65 VTE recurrences for every 1000 person years of follow-up), and the conservative value of R^{2}_{cs} (0.051), with all chosen values based on Ensor et al.46 Now criteria T1 to T3 can be checked, for example by typing in Stata:

This indicates that at least 5143 participants are required, corresponding to 692 events and an EPP of 23.1. This is considerably more than 10 EPP, and is driven by a desired shrinkage factor (criterion T2) of only 10% to minimise overfitting based on just 8% of variation explained by the model. If the number of candidate predictor parameters is lowered to 20, the required sample size is reduced to 3429 (still an EPP of 23.1).

### Example 3: Continuous outcome

Hudda et al developed a prediction model for fat free mass in children and adolescents aged 4 to 15 years, including 10 predictor parameters based on height, weight, age, sex, and ethnicity.47 The model is needed to provide an estimate of an individual’s current fat mass (=weight minus predicted fat free mass). On external validation, the model had an R^{2}_{cs} of 0.90. Let us assume that the model will need updating (eg, in 10 years owing to changes in the population behaviour and environment), and that an additional 10 predictor parameters (and thus a total of 20 parameters) will need to be considered in the model development.

The sample size for a model development dataset must at least meet the four criteria of C1 to C4 in box 1. This requires us to specify the anticipated R^{2}_{cs} (0.90), number of candidate predictor parameters (n=20), and mean (26.7 kg) and standard deviation (8.7 kg) of fat free mass in the target population (taken from Hudda et al47). For example, in Stata, after installation of pmsampsize (type: ssc install pmsampsize), we can type:

This returns that at least 254 participants are required, and so 12.7 participants for each predictor parameter. The sample size of 254 is driven by the number needed to precisely estimate the model standard deviation (criterion C3), as only 68 participants are needed to minimise overfitting (criteria C1 and C2).

## Extensions and further topics

### Ensuring accurate predictions in key subgroups

Alongside the criteria outlined in box 1, a more stringent task is to ensure model predictions are accurate in key subgroups defined by particular values or categories of included predictors.48 One way to tackle this is to ensure predictor effects in the model equation are precisely estimated, at least for key subgroups of interest.1516 For binary and time-to-event outcomes, the precision of a predictor’s effect depends on its magnitude, the variance of the predictor’s values, the predictor’s correlation with other predictors in the model, the sample size, and the outcome proportion or rate in the study.495051 For continuous outcomes, it depends on the sample size, the residual variance, the correlation of the predictor with other included predictors, and the variance of the predictor’s values.4852535455 Note that for important categorical predictors large sample sizes might be needed to avoid separation issues (ie, where no events or non-events occur in some categories),13 and potential bias from sparse events.56

### Sample size considerations when using an existing dataset

Our proposed sample size calculations (ie, based on the criteria in box 1) are still useful in situations when an existing dataset is already available, with a specific number of participants and predictors. Firstly, the calculations might identify that the dataset is too small (for example, if the overall outcome risk cannot be estimated precisely) and so the collection of further data is required.5758 Secondly, the calculations might help identify how many predictors can be considered before overfitting becomes a concern. The shrinkage estimate obtained from fitting the full model (including all predictors) can be used to gauge whether the number of predictors could be reduced through data reduction techniques such as principal components analysis.10 This process should be done blind to the estimated predictor effects in the full model, as otherwise decisions about predictor inclusion will be influenced by a “quick look” at the results (which increases the overfitting).

### Sample size requirements when using variable selection

Further research on sample size requirements with variable selection is required, especially for the use of more modern penalisation methods such as the lasso (least absolute shrinkage and selection operator) or elastic net.3359 Such methods allow shrinkage and variable selection to operate simultaneously, and they even allow the consideration of more predictor parameters than number of participants or outcome events (ie, in high dimensional settings). However, there is no guarantee such models solve the problem of overfitting in the dataset at hand. As mentioned, they require penalty and shrinkage factors to be estimated using the development dataset, and such estimates will often be hugely imprecise. Also, the subset of included predictors might be highly unstable60616263; that is, if the prediction model development was repeated on a different sample of the same size, a different subset of predictors might be selected and important predictors missed (especially if sample size is small). In healthcare the final set of predictors is a crucial consideration, owing to their cost, time, burden (eg, blood test, invasiveness), and measurement requirements.

### Larger sample sizes might be needed when using machine learning approaches to develop risk prediction models

An alternative to regression based prediction models are those based on machine learning methods, such as random forests and neural networks (of which “deep learning” methods are a special case).64 When the focus is on individualised outcome risk prediction, it has been shown that extremely large datasets might be needed for machine learning techniques. For binary outcomes, machine learning techniques could need more than 10 times as many events for each predictor to achieve a small amount of overfitting compared with classic modelling techniques such as logistic regression, and might show instability and a high optimism even with more than 200 EPP.26 A major cause of this problem is that the number of predictor (“feature”) parameters considered by machine learning approaches will usually far exceed that for regression, even when the same set of predictors is considered, particularly because they routinely examine multiple interaction terms and categorise continuous predictors.

Therefore, machine learning methods are not immune to sample size requirements, and actually might need truly “big data” to ensure their developed models have small overfitting, and for their potential advantages (eg, dealing with highly non-linear relations and complex interactions) to reach fruition. The size of most medical research datasets is better suited to using regression (including penalisation and shrinkage approaches),65 especially as regression also leads to a transparent model equation that facilitates implementation, validation, and graphical displays.

## Sample size for model updating

When an existing prediction model is updated, the existing model equation is revised using a new dataset. The required sample size for this dataset depends on how the model is to be updated and whether additional predictors are to be included. In our worked examples, we assumed that all parameters in the existing model will be re-estimated using the model updating dataset. In that situation, the researcher can still follow the guidance in box 1 for calculating the required sample size, with the total predictor parameters the same as in the original model plus those new parameters required for any additional predictors.

Sometimes, however, only a subset of the existing model’s parameters is to be updated.6667 In particular, to deal with calibration-in-the-large, researchers might only want to revise the model intercept (or baseline survival), while constraining the other parameter estimates to be the same as those in the existing model. In this case the required sample size only needs to be large enough to estimate the mean outcome value or outcome risk precisely (ie, to meet criteria C1, B1, or T1 in box 1). Even if researchers also want to update the existing predictor effects, they might decide to constrain their updated values to be equal to the original values multiplied by a constant. Then, the sample size only needs to be large enough to estimate one predictor parameter (ie, the constant) for the existing predictors, plus any new parameters the researchers decide to add. Such model updating techniques therefore reduce the sample size needed (to meet the criteria in box 1) compared with when every predictor parameter is re-estimated without constraint.

## Conclusion

Patients and healthcare professionals require clinical prediction models to accurately guide healthcare decisions.1 Larger sample sizes lead to more robust models being developed, and our guidance in box 1 outlines how to calculate the minimum sample size required. Clearly, the more data for model development the better; so if larger sample sizes are achievable than our guidance suggests, use it! Of course, any data collected should be of sufficient quality and representative of the target population and settings of application.6869

After data collection, careful model building is required using appropriate methods.1310 In particular, we do not recommend data splitting (eg, into model training and testing samples), as this is inefficient and it is better to use all the data for model development, with resampling methods (such as bootstrapping) used for internal validation.7071 Sometimes external information might be used to supplement the development dataset further.727374 Lastly, sample size requirements when externally validating an existing prediction model require a different approach, as discussed elsewhere.75767778

## Footnotes

Contributors: RDR and MVS led the methodology that underpins the methods in this article, with contributions from all authors. RDR wrote the first and updated drafts of the article, with important contributions and revisions from all authors at multiple stages. JE led the development of the pmsampsize packages in Stata and R. RDR is the guarantor.

Funding: KGMM receives funding from the Netherlands Organisation for Scientific Research (project 9120.8004 and 918.10.615). For his work on this paper, FEH was supported by CTSA award No UL1 TR002243 from the National Center for Advancing Translational Sciences. Its contents are solely the responsibility of the authors and do not necessarily represent official views of the National Center for Advancing Translational Sciences or the US National Institutes of Health. GC is supported by the National Institute for Health Research (NIHR) Biomedical Research Centre, Oxford. KIES is funded by the NIHR School for Primary Care Research. This (publication/paper/report) presents independent research funded by the NIHR. The views expressed are those of the authors and not necessarily those of the National Health Service, NIHR, or Department of Health and Social Care.

Competing interests: We have read and understood the BMJ Group policy on declaration of interests and declare we have no competing interests.

Provenance and peer review: Not commissioned; peer reviewed.

Patient and public involvement: Patients or the public were not involved in the design, conduct, reporting, or dissemination of our research.

Dissemination to participants and related patient and public communities: We plan to disseminate the sample size calculations to our Patient and Public Involvement and Engagement team when they are applied in new research projects.

The lead author (RDR) affirms that the manuscript is an honest, accurate, and transparent account of the study being reported; that no important aspects of the study have been omitted; and that any discrepancies from the study as planned (and, if relevant, registered) have been explained.