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Estimation of Prostate Cancer Probability by Logistic Regression: Free and Total Prostate-specific Antigen, Digital Rectal Examination, and Heredity Are Significant Variables

Arja Virtanen^1,a, Mehran Gomari², Ries Kranse³ and Ulf-Hkan Stenman⁴

¹ Central Laboratory, University Central Hospital of Turku, Medical Informatics Research Centre in Turku (MIRCIT), FIN-20521 Turku, Finland.

² Turku Centre for Computer Science (TUCS), FIN-20520 Turku, Finland.

³ Department of Urology, Dijkzigt Academic Hospital Rotterdam, NL-3015 GD Rotterdam, The Netherlands.

⁴ Department of Clinical Chemistry, University Central Hospital of Helsinki, FIN-00100 Helsinki, Finland.
^a Author for correspondence. Fax 358-2-2613920; e-mail arja.virtanen{at}utu.fi

	Abstract

Top Abstract Introduction Materials and Methods Results Discussion References

 
Background: Despite low specificity, serum prostate-specific antigen (PSA) is widely used in screening for prostate cancer. Specificity can be improved by measuring free and total PSA and by combining these results with clinical findings. Methods such as neural networks and logistic regression are alternatives to multistep algorithms for clinical use of the combined findings.

Methods: We compared multilayer perceptron (MLP) and logistic regression (LR) analysis for predicting prostate cancer in a screening population of 974 men, ages 55–66 years. The study sample comprised men with PSA values >3 µg/L. Explanatory variables considered were age, free and total PSA and their ratio, digital rectal examination (DRE), transrectal ultrasonography, and a family history of prostate cancer.

Results: When at least 90% sensitivity in the training sets was required, the mean sensitivity and specificity obtained were 87% and 41% with LR and 85% and 26% with MLP, respectively. The cancer specificity of an LR model comprising the proportion of free to total PSA, DRE, and heredity as explanatory variables was significantly better than that of total PSA and the proportion of free to total PSA (P <0.01, McNemar test). The proportion of free to total PSA, DRE, and heredity were used to prepare cancer probability curves.

Conclusion: The probability calculated by logistic regression provides better diagnostic accuracy for prostate cancer than the presently used multistep algorithms for estimation of the need to perform biopsy.© 1999 American Association for Clinical Chemistry

	Introduction

Top Abstract Introduction Materials and Methods Results Discussion References

 
Prostate cancer is the most common cancer in men, and it is the second most common cause of cancer death (1). When localized to the prostate, the cancer can usually be cured by radical prostatectomy. Therefore, attempts are being made to improve the prognosis by early detection and screening (2)(3). Prostate-specific antigen (PSA)¹ is a sensitive marker for prostate cancer (4)(5). When PSA is used as a primary test for screening, ~10% of all men >50 years of age have a PSA value exceeding the commonly used cutoff of 4 µg/L (2)(3). A majority of patients with increased PSA have benign prostatic hyperplasia (2)(3). Prostate cancer is found in approximately one-third of the cases by the use of digital rectal examination (DRE), transrectal ultrasonography (TRUS), and sextant biopsy.

Because the examinations required to confirm the diagnosis are expensive, several approaches are being used to increase the accuracy of the primary screening method, e.g., the PSA value divided by the prostate volume (PSA density) (6), the rate of increase of serum PSA (PSA velocity) (7), and the proportion of the two major forms of PSA in serum, i.e., free PSA and the complex of PSA with ₁-antichymotrypsin. In patients with prostate cancer, the proportion of PSA-₁-antichymotrypsin is higher and that of free PSA lower than in those with benign prostatic hyperplasia (8). If the complexed, total free, and total PSA are measured, the number of false-positive results can be reduced by 30–50%, but a further improvement is desirable (8)(9)(10).

We have evaluated whether the diagnostic accuracy can be improved by utilizing all the diagnostic information available to determine whether a biopsy should be performed. Presently, the most common algorithm is to perform biopsies on all men with a PSA value >4 µg/L or a positive DRE in combination with a lower PSA value. ROC curves and logistic regression analysis have been used to evaluate the diagnostic methods for prostate cancer. Neural networks may be used as alternatives to statistical methods for medical decision support (11), and recently artificial neural networks have also been utilized (12). However, comparisons with standard statistical analysis are lacking.

In this study, we evaluated the ability of logistic regression (LR) and multilayer perceptron (MLP) to improve the diagnostic accuracy for prostate cancer by combining determinations of free and total PSA, DRE, TRUS, age, and a family history of prostate cancer.

A logistic regression model can be formulated mathematically by relating the probability of some event conditional on the vector x of explanatory variables (13). The linear logistic model is described by the formula:

(1)

where P_x is the probability that the response belongs in the cancer group when explanatory variables have certain values. In the equation above, is the intercept parameter and ß is the vector of the slope parameters, which are estimated from the data. For interpretation of the results, the slope parameters have an important meaning. Odds ratios can be calculated by taking the antilogarithm of these values. If the slope parameter is positive, the odds ratio describes how much the odds for having a cancer increase when the corresponding explanatory variable increases one unit and the other variables remain unchanged. A negative slope parameter correspondingly decreases the odds. Some of the variables may be useful for the classification, but when combined with other variables, they may be ineffective if they are highly correlated with other variables. Variables that contribute redundant information can be omitted, with only those variables with the best diagnostic accuracy retained in the model. In the present study, the probability of cancer is modeled, and a stepwise variable selection method is used to provide an adequate solution to the problem of variable selection (14).

Artificial neural networks (ANNs) form a family of computational architectures inspired by biological brains [for a review, see Ref. (15)]. ANNs can approximate nonlinear functions using only the input-output patterns in a supervised learning task. The knowledge in ANNs is distributed among artificial neurons. Therefore, they can be taught to approximate functions with incomplete or noisy data. In this study we focus our attention on MLPs, which are feed-forward ANNs.

MLPs are simple networks with successive layers containing different numbers of "units" ("neurons" or "nodes").The units are the basic processing elements of the network, and they are connected to each other by "connections" ("links" or "arcs"). A numeric "weight" is associated with each connection. Units with the same task form a layer. An MLP consists of three different types of layers: input, output, and hidden (layers between the input and output layers) layers. The units in each layer receive their inputs from the units in the previous layer (feed-forward), and there are no feedback loops. Each unit has an activation function for computing the activation level and a threshold (bias) for defining the current activation level.

Learning in MLPs takes place by adjusting the weights between layers so that the difference between the actual (computed by the MLP) and the desired output is minimized. The learning algorithm used for this purpose is usually "backpropagation" (16), and a typical error function to be minimized is the mean square error.

	Materials and Methods

Top Abstract Introduction Materials and Methods Results Discussion References

 
subjects
The data used in this study were extracted from a prostate cancer screening database comprising 974 men in Rotterdam, The Netherlands (17). Input variables were total and free PSA, the proportion of free to total PSA (F/T ratio), DRE, and TRUS, as well as age and family history of prostate cancer. The outcome, i.e., prostate cancer or no prostate cancer, was based on histological examination of sextant prostate biopsies. Blood specimens were collected before DRE and TRUS were performed. Before TRUS, DRE was performed by different well-trained urology residents. Prostate nodules, asymmetry of the gland, induration, and bogginess were considered abnormal. A suspicious TRUS was defined as a hypoechoic lesion. DRE and TRUS were performed without knowledge of the other tests, and they were performed by a different investigator. The family history was obtained from the baseline questionnaire that was filled out by every participant and returned together with the signed informed consent form. A positive or negative family history of prostate cancer included only first-degree male relatives. From this database, all subjects 55–66 years of age with PSA concentrations in the diagnostic gray zone between 3 and 10 µg/L were selected for the study (n = 212). Among men with total PSA <3 µg/L, the proportion of cancer was 7.4%, and in those with total PSA >10 µg/L, it was 65.3%. In the study sample, the proportion of cancer was 25%. Another study group comprised all men 55–66 years of age with a PSA 3 µg/L (n = 241).

psa determinations
Total- and free-PSA serum samples were measured with the ProStatus PSA Free/Total assay (EG&G Wallac, Turku, Finland). This assay provides simultaneous measurement of free and total PSA by the use of time-resolved DELFIA immunofluorometry. Assay performance characteristics have been reported previously (17)(18).

methods
To estimate the performance of the different methods, a fivefold "cross-validation" was used. The study sample was divided randomly into five equal subsamples. One subsample at a time formed a test set (~20% of the study sample), and the remaining four subsamples together formed the training set. This protocol was repeated five times.

LR analysis.
LR can be used when the response is binary (e.g., cancer/non-cancer). To investigate the effect of explanatory variables to predict prostate cancer, we used a stepwise selection method. In this method, the sequences of regression equations are computed, and at each step an explanatory variable is automatically added to or deleted from the model, depending on the statistical significance of the variable. To describe the association between response and explanatory variables, an adjusted coefficient of determination (Nagelkerke's generalized R²) and odds ratios with 95% profile likelihood confidence intervals were evaluated. For identification of extreme values, we used the regression diagnostics developed by Pregibon (19). The adequacy of the fitted model was further checked by the Hosmer-Lemeshow test (20). Probability curves for cancer were evaluated by LR analysis. Both the training and validation sets were used for model building. All statistical calculations were performed with SAS^® for Windows, Ver. 6.11 (SAS Institute), and graphic presentation with Microcal Origin^TM for Windows (Microcal Software).

MLP.
A faster version of the backpropagation algorithm—resilient backpropagation (RPROP)—was used (21). Several MLPs with different numbers of hidden units ranging from 2 to 20 in one hidden layer were examined. For all MLPs, the activation function of the input units was "identity", whereas a "symmetric logistic" activation function was used for the hidden and output units. To avoid "overfitting" of the MLPs to the training sets, we used the "early stopping" method (22). This means that part of the data was used as a validation set (for each cross-validation trial) and the training phase was terminated when the error of the validation set was at a minimum. This was achieved by the use of a batch program that always updated a record of the network status with the minimum error of the validation set during the training process. Because the early stopping method is sensitive to the initial values of Neural Network weights, 10 different random initial weights were used for each cross-validation trial, and the best one was chosen according to the validation set. For the MLP experiments, the Stuttgart Neural Network Simulator (SNNS), Ver. 4.1, was used.

statistical methods
The association between categorical variables and diagnostic groups was tested by ² statistics, and the differences in numerical variables were tested by Wilcoxon statistics. Sensitivity, specificity, efficiency, and positive and negative predictive values were calculated to compare different methods. Because sensitivity for the detection of cancer was considered more important than specificity, the threshold was chosen so that sensitivity was at least 90% in all training sets (validation sets for the MLPs). Furthermore, the efficiency had to be at its maximum. If the same efficiency was obtained with different combinations of sensitivity and specificity, the threshold with the highest sum of sensitivity and specificity was chosen. After the best threshold values from the training or validation sets were determined, we used the values in test sets to evaluate the performance of the different methods. Statistical comparisons between the methods were calculated with the multivariate Wilks lambda statistics (23). To compare the classification results between the methods and with the total PSA and the F/T ratio on the test sets, we calculated pairwise comparisons using the McNemar test (24) with Bonferroni correction.

	Results

Top Abstract Introduction Materials and Methods Results Discussion References

 
The descriptive statistics, frequency distributions, and probability for Wilcoxon and ² statistics for the study sample are shown in Table 1 . Of the 212 men, 53 (25%) had prostate cancer. The proportions of abnormal results for the categorical variables DRE, TRUS, and heredity were 8.8%, 13.2%, and 8.2% in the non-cancer group and 41.5%, 41.5%, and 22.6% in the cancer group, respectively. There was a statistically significant association between the diagnostic group and categorical variables (all P <0.01). There was a significant association between TRUS and DRE (P = 0.001) and a moderate association between TRUS and heredity (P = 0.037), but not between DRE and heredity (P = 0.118). There was no difference between the groups with respect to age (P = 0.752). The median F/T ratio in the non-cancer group was 0.18 vs 0.12 in the cancer group (P <0.001).

lr
In the stepwise variable selection method, age and free PSA were not statistically significant variables in any of the training sets. The F/T ratio and DRE were nearly always added to the model in the first or second step, and TRUS and heredity were added in the third or fourth steps. The F/T ratio, DRE, and heredity were chosen for the final models. The aptness of these models was checked, and one observation in each of the four training sets was removed because it was poorly elucidated by the model (19). The slope parameters with significance levels, odds ratios with 95% profile likelihood confidence intervals, and adjusted coefficient of determination (R²) in five different training sets obtained with the final models are shown in Table 2 . In all training sets, DRE was statistically significant at the 1% level. The F/T ratio was significant at the 1% level except in one training set, and heredity was significant at the 5% level except in one training set. The slope parameters for DRE and heredity were positive, and those for the F/T ratio were negative. A change from a normal DRE value (0) to an abnormal value (1) increased the cancer risk approximately sixfold, and a positive heredity increased the risk approximately threefold. The confidence intervals for these odds ratios were wide, reflecting the small sample size. However, in all training sets, the risk was at least 1.7-fold when DRE was abnormal.

An increased cancer risk was associated with low F/T ratios. When the F/T ratio decreased by 0.12 units, the cancer risk increased, on average, fivefold. However, for clarity, the odds ratios in Table 2 display the decrease in the risk when the F/T ratio increased by 0.12 units.

Using the calculated parameter estimates obtained by LR, we could determine the probability that a man has a prostate cancer with certain values for the predictive variables (e.g., F/T ratio, DRE, and heredity). In Fig. 1 , probability curves for cancer associated with different values and combinations of the F/T ratio, DRE, and heredity are plotted for PSA values between 3 and 10 µg/L. These results clearly demonstrate the importance of the F/T ratio.

View larger version (20K): [in this window] [in a new window]   Figure 1. Probability of cancer derived from LR with different results for DRE (Dre), heredity (0 = normal, 1 = abnormal), and the F/T ratio for PSA concentrations of 3–10 µg/L.

When LR is performed for total-PSA values between 3 and 45 µg/L with the combination of total PSA, the F/T ratio, and DRE as explanatory variables, the area under the ROC curve was 0.809 and R² was 22%. The greatest area under the curve (0.812) was achieved when total PSA and the F/T ratio were combined with DRE, TRUS, and heredity. Because the difference between the models was not critical, the cancer probability curves plotted in Fig. 2 were determined from the model in which total PSA concentrations of 3–45 µg/L, the F/T ratio, and DRE are included.

View larger version (21K): [in this window] [in a new window]   Figure 2. Probability of cancer derived from LR of total PSA (µg/L), the F/T ratio, and a normal DRE (A) or abnormal DRE (B) for PSA concentrations of 3–45 µg/L.

mlp
For the MLP experiments, all other explanatory variables except age were selected as input variables. The final thresholds for the MLPs were chosen on the basis of the performance of the model in the validation sets instead of the training sets because the termination of the training was based on the validation sets and not on the training sets. Among the MLPs examined, the MLP with five hidden units provided the best performance on unseen data.

comparisons of the methods
The performance measures of the different methods are shown in Table 3 . The mean differences between the methods were not statistically significant when all performance measures were considered (P = 0.154, Wilks lambda test). However, a statistically significant difference (P = 0.023, Wilks lambda test) between the methods was observed when sensitivity and specificity were considered. This difference was attributed to the fact that the specificity of the MLP was only approximately one-half of that of the LR method. The LR model was significantly better in predicting prostate cancer than the MLP or the model with total PSA and the F/T ratio as explanatory variables (P <0.01, McNemar test).

	Discussion

Top Abstract Introduction Materials and Methods Results Discussion References

 
We examined the ability of two methods, LR and MLP, to improve the diagnostic accuracy for prostate cancer in men 55–66 years of age with PSA concentrations of 3–10 µg/L. As used, the MLP did not provide information about the importance of the different variables. However, when LR was used, the F/T ratio, DRE, and heredity were found to be independent predictors of prostate cancer, whereas age, and total and free PSA were not. With the explanatory variables selected, LR predicted prostate cancer significantly better than the MLP and the combination of total PSA and the F/T ratio alone (P <0.01, McNemar test). However, the impact of total PSA is artificially restricted by the use of a limited concentration range, and when all cases with PSA concentrations of 3–45 µg/L were considered, both total PSA and the F/T ratio were significant variables.

Of the explanatory variables examined, the F/T ratio and DRE were the most significant predictors of prostate cancer in the restricted PSA concentration range 3–10 µg/L. This is in agreement with earlier studies (17)(25)(26)(27)(28)(29)(30). Heredity also improved the diagnostic accuracy. The effect of age was not significant, probably because of the narrow age distribution, only 10 years, whereas it was nearly 40 years in the study of Chen et al. (26). Free PSA was not an independent variable in LR, apparently because it contained repetitive information already included in the F/T ratio.

DRE and TRUS were more sensitive (41.5% each) than heredity (22.6%), but the stepwise method selected only DRE to the model, apparently because it was slightly more specific (91.2%) than TRUS (86.8%) and because of considerable covariance between these variables. This finding was obtained although different examiners performed the examinations without knowledge of the result of the other test. TRUS was a significant variable in only one training set and did not add diagnostic information to the selected logistic model. However, in an LR model where TRUS replaced DRE, TRUS was statistically significant in every training set. A positive result in TRUS increased the cancer risk 3.5-fold, whereas the corresponding increase for DRE was sixfold. Therefore DRE was included in the calculations of probability curves. In this study, TRUS did not add statistical power and, therefore, was not included.

The results obtained with TRUS and DRE are highly observer dependent; therefore, our results may not be reproducible in a different setting. TRUS and DRE generally are thought to provide independent information, but in a study where this was evaluated by LR, only PSA, DRE, and PSA related to prostate volume were found to predict the presence of prostate cancer (31).

In a screening setting, PSA often is determined first, and DRE is performed only on patients with abnormal results. In a clinical setting, this information is also usually available before the decision to perform a biopsy is made. Presently, the F/T ratio mainly is used to evaluate the need for a biopsy if total PSA is 4–10 µg/L, whereas biopsy is always considered as indicated if PSA is >10 µg/L. The probability curves shown in Figs. 1 and 2 suggest that this approach is fairly crude. The mean cancer probability associated with PSA values of 4–10 µg/L is ~25%. Fig. 2A shows that the risk is ~15% when PSA is 4 µg/L, the F/T ratio is ~0.20, and DRE is normal. Fig. 2A also demonstrates how strongly changes in the F/T ratio affect the risk. Thus, a F/T ratio of 0.4 in combination with a total-PSA concentration of 30 µg/L is also associated with a cancer probability of ~15%. This shows that the F/T ratio could be utilized at much higher PSA concentrations than are used at present. A low F/T ratio increases the risk very strongly. Thus, a total-PSA concentration of 3 µg/L and a F/T ratio of 0.1 are associated with a 30% risk. This shows that the trend to lower the cutoff for total PSA from 4 to 3 µg/L (32)(33) is justified if the F/T ratio is used to reduce the number of false-positive results. Fig. 2B shows that an abnormal DRE causes a considerable further increase in cancer risk, and biopsy is indicated in practically all men with PSA >3 µg/L and positive DRE regardless of the F/T ratio. The risk is further increased by a positive heredity (Fig. 1 ), and the combination of positive DRE and heredity indicates a need for biopsy for all patients with a PSA concentration >3 µg/L. The combined risk of a positive heredity and a PSA concentration <3 µg/L deserves to be investigated.

As shown in Figs. 1 and 2 , the additional impact of DRE and heredity on prostate cancer probability is considerable compared with total PSA and the F/T ratio. However, there is room for further improvement in diagnostic accuracy, and with the statistical tools used in the present study, other variables can easily be included. It will be interesting to see whether utilization of PSA density, PSA velocity, and prostate volume provides further independent diagnostic information (6)(7)(29). Presenting the probabilities graphically becomes impractical when more than three variable are included. However, with LR, the cancer probability can be expressed with a single value, which could replace the multistep algorithms used at present.

Probability curves and tables for a continuous range of F/T ratios based on published cancer prevalence rates have been presented recently by Marley et al. (34), but to our knowledge, curves based on the combined impact of all the diagnostic variable used in the present study have not been presented before. When comparing our results with those of Marley et al. (34), it should be noted that our curves are based on the study sample and that cancer prevalence rates have not been included in the calculations.

The correlations between the ProStatus assays and some other widely used assays, the Abbott IMx and the Hybritech Tandem E assays, are very good (35)(36), but it still is advisable to establish risk algorithms separately for each different assay. Many other assays give quite different results, and for these the algorithms established are not applicable. It is also necessary to consider racial differences when estimating prostate cancer risk on the basis of PSA values (37). The impact of heredity also needs to be evaluated separately in various populations.

The validity at 90% sensitivity is important, and the corresponding cutoff is clinically relevant because it facilitates detection of prostate cancer at a potentially curable stage.

One of the aims of the present study was to compare the performance of LR and MLP. Our results suggest that LR was slightly better, but this may not be a generally valid conclusion because of the limited number of cases studied. Smaller training sets had to be used than in LR because part of each training set was used as the validation set in MLP. Furthermore, only a fivefold cross-validation could be used. In a recent study by Gomari et al. (38) comprising a larger number of cases, the performance of LR and MLP was equal.

In conclusion, the probability of prostate cancer can be more accurately estimated by the use of LR with multiple explanatory variables than by the use of only total PSA and the F/T ratio. DRE and heredity were found to be independent variables that substantially affected cancer probability. The results of TRUS added no statistical power to predict prostate cancer over the findings of DRE alone, and hence were not included as an explanatory variable in either LR model. For patients with PSA concentrations between 3–10 µg/L, LR analysis (using the F/T ratio, the DRE examination, and heredity) allows calculation of a single value for the probability of finding prostate cancer on biopsy. This method may offer advantages over the multistep algorithms presently used to determine the need to perform prostate biopsy.

	Acknowledgments

 
This work was supported by Tekes (Technology Development Centre Finland) and EG&G Wallac. We thank Jari Forsström, Veli Kairisto, Esa Uusipaikka, and Timo Järvi for helpful comments and discussions.

	Footnotes

 
¹ Nonstandard abbreviations: PSA, prostate-specific antigen; DRE, digital rectal examination; TRUS, transrectal ultrasonography; LR, logistic regression; MLP, multilayer perceptron; ANN, artificial neural network; and F/T ratio, proportion of free to total PSA.

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