This article has Open Peer Review reports available.

# Explaining computation of predictive values: 2 × 2 table versus frequency tree. A randomized controlled trial [ISRCTN74278823]

- Anke Steckelberg
^{1}Email author, - Andrea Balgenorth
^{1}, - Jürgen Berger
^{2}and - Ingrid Mühlhauser
^{1}

**4**:13

https://doi.org/10.1186/1472-6920-4-13

© Steckelberg et al; licensee BioMed Central Ltd. 2004

**Received: **28 February 2004

**Accepted: **10 August 2004

**Published: **10 August 2004

## Abstract

### Background

Involving patients in decision making on diagnostic procedures requires a basic level of statistical thinking. However, innumeracy is prevalent even among physicians. In medical teaching the 2 × 2 table is widely used as a visual help for computations whereas in psychology the frequency tree is favoured. We assumed that the 2 × 2 table is more suitable to support computations of predictive values.

### Methods

184 students without prior statistical training were randomised either to a step-by-step self-learning tutorial using the 2 × 2 table (n = 94) or the frequency tree (n = 90). During the training session students were instructed by two sample tasks and a total of five positive predictive values had to be computed. During a follow-up session 4 weeks later participants had to compute 5 different tasks of comparable degree of difficulty without having the tutorial instructions at their disposal. The primary outcome was the correct solution of the tasks.

### Results

There were no statistically significant differences between the two groups. About 58% achieved correct solutions in 4–5 tasks following the training session and 26% in the follow-up examination.

### Conclusions

These findings do not support the hypothesis that the 2 × 2 table is more valuable to facilitate the calculation of positive predictive values than the frequency tree.

## Background

Diagnostic procedures are increasingly expected by consumers to ensure their health; "certainty" has become a product [1]. Assuming that test results are certain, only a minority is aware about false positive and false negative alarms. Previous research has shown that even physicians have great difficulties in estimating the positive predictive values of diagnostic tests [2–4]. One study reported that 95 out of 100 physicians estimated the positive predictive value of screening mammography to be between 70–80% rather than 7.8% [2]. Similar results were reported for AIDS counselors for low-risk clients. The majority of counselors assured that false positives would never occur and half of the counselors incorrectly assured that if a low-risk person tests positive, it is absolutely certain (100%) that he or she is infected with the virus [5]. An incorrect probability judgment may result in unnecessary tests or pseudo certainty. Therefore, the understanding, presentation and communication of test quality are a challenge for both: lay people and professionals.

Involving lay people in decision making on diagnostic procedures requires a basic level of statistical thinking. Help for computing Bayesian inference is needed. Statistical thinking can be enhanced by representing statistical information in terms of natural frequencies rather than probabilities [6, 7]. This is explained by the evolution of the human reasoning system. Gigerenzer proposed that human reasoning is algorithms designed for information that comes in a format that was present in the "environment of evolutionary adaptiveness" [8]. Human reasoning processes are adapted to natural frequencies. Also Bayesian computations are easier when the information is communicated this way.

In the present study, we compare the two visual helps in non-medical students. We hypothesized that the 2 × 2 table is more eligible than the frequency tree to facilitate correct answers in tasks of calculations of positive predictive values 4 weeks after an initial training-session. We also describe students' ability to calculate positive predictive values, analyzing the transfer of the numerical information into the visual help and the correct computation.

## Methods

### Participants

We approached 238 students without prior statistical training to recruit the necessary 184 students who agreed to participate. (See power calculation below) Students attending the University of Hamburg (health sciences, biology and sports), a vocational college (health and nursing) or taking part in an in-service training (nursing and public health) were informed about the timing and content procedure of the study during their courses.

### Procedure

The study was carried out between October 2000 and July 2001 and consisted of two supervised sessions lasting about 1 h each. The recruited 184 students were randomly assigned either to the frequency tree group (n = 94) or to the 2 × 2 table group (n = 90) using blocked randomization in blocks of 10. Concealed allocation based on computer-generated random numbers was done by an external person. In addition, the external person prepared sealed envelopes for both sessions including the tutorial with the tasks and a questionnaire for survey of age, gender, years of school, mark in mathematics and social state. The training consisted of a written step-by-step self-learning tutorial (Additional file 1, 2, 3). The participants had to compute 5 positive predictive values in each session. The tutorial and tasks followed the recommendations for the presentation of numerical information [4]. Participants were asked to reveal how they achieved their solutions. Participants were allowed to use a pocket calculator. Correct results were presented and discussed after each session.

In the follow-up examination participants were again asked to solve 5 different diagnostic problems of similar level of difficulty but without having the tutorial instructions at their disposal (Additional file 4,5,6). Participants who missed the date were repeatedly contacted by letter, phone or e-mail. Efforts were discontinued after 4 weeks.

### Assessing performance

#### Correct solution of the tasks

A solution was classified correct, when the documented positive predictive value was equivalent to the correct solution rounding up or down to the next full percentage point. If a participant used the correct computation (correct positives divided by all positives) but made a calculation error either in the transfer of the numerical information into the visual help or within the division, we ignored calculation errors. Whenever a different computation such as rule of three – a mechanical method for solving proportions – was used or the calculation protocol was missing the rounded solutions were classified likewise as correct by congruence. If the protocol indeed showed that a correct rounded solution resulted from an incorrect computation such as positive predictive value = correct positives / false positives the answer was classified as incorrect. Tasks that had not been worked on were also classified incorrect.

#### Correct transfer

To evaluate the usefulness of the different visual helps, we evaluated the ability of correct transfer of the numerical information into the charts. A transfer was classified as correct, when the numerical information of the problems was inserted into the gaps provided. It was sufficient to insert the relevant values for the computation, calculation errors were ignored.

#### Correct computation

The computation was classified as correct Bayesian approach when the following computation was used: positive predictive value = correct positives / (correct positives + false positives) or positive predictive value = correct positives / all positives. The computation was classified as Non-Bayesian approach when the computation was used with false values. Other computations were classified as other strategies.

### Statistical power and analyses

Hypothesized distribution of correct answers after 4 weeks between the two study groups

Categories* (numbers of correct answers) | Frequency tree | 2 × 2 table |
---|---|---|

0 | 0.40 | 0.30 |

1 | 0.15 | 0.05 |

2 | 0.15 | 0.05 |

3 | 0.10 | 0.20 |

4 | 0.10 | 0.20 |

5 | 0.10 | 0.20 |

Analysis is based on the intention-to-participate principle that includes all randomised participants as randomised. Drop outs were considered as having solved none of the positive predictive values correctly.

## Results

Baseline characteristics*

Age | ||
---|---|---|

Frequency tree (n = 94) | 2 × 2 table (n = 90) | |

Median (range) | 29 (20–54) | 26 (19–51) |

Missing values | 3 (3) | 2 (2) |

| ||

Male | 15 (16) | 20 (22) |

Female | 77 (82) | 67 (75) |

Missing values | 2 (3) | 3 (3) |

| ||

< 10 years | 1 (1) | 1 (1) |

10–12 years | 22 (23) | 19 (21) |

> 12 years | 68 (72) | 67 (75) |

Missing values | 3 (3) | 3 (3) |

| ||

1 (highest level) | 6 (6) | 6 (7) |

2 | 20 (21) | 18 (20) |

3 | 35 (37) | 32 (36) |

4 | 14 (15) | 17 (19) |

5 (lowest level) | 5 (5) | 9 (10) |

Missing values | 14 (15) | 8 (9) |

| ||

University of Hamburg | 59 (63) | 55 (61) |

Vocational College | 14 (15) | 15 (17) |

Non-academic students | 21 (22) | 20 (22) |

### Correct solutions of the tasks

Numbers of correct solutions of positive predictive values*

Category | Training session | Follow-up examination | ||
---|---|---|---|---|

Frequency tree (n = 94) | 2 × 2 table (n = 90) | Frequency tree (n = 74) | 2 × 2 table (n = 75) | |

(0–1 answer correct) | 19(20) | 18(20) | 53 (72) | 50 (67) |

(2–3 answers correct) | 20 (21) | 20 (22) | 2 (3) | 5 (7) |

(4–5 answers correct) | 55 (59) | 52(58) | 19 (26) | 20(27) |

Within the category 4–5 correct answers 27% of participants (95% CI, 17%–38%) (2 × 2 table) and 26% (95% CI, 16%–37%) (frequency tree) had correct solutions. The differences between the two study groups were not statistically significant neither in the training session (p = 0.95 {0.49 one-sided}) nor in the follow-up examination (p = 0.48 {0.24} for the analysis on intention-to-participate and p = 0.61 {0.31} for the analysis on-participation (Table 3).

Analysis of each task regarding correct solutions, transfer of numerical information and Bayesian computations*

Correct solution | ||||
---|---|---|---|---|

Training session | Follow-up examination | |||

frequency tree (n = 94) | 2 × 2 table (n = 90) | frequency tree (n = 74) | 2 × 2 table (n = 75) | |

Task A | 67 (71) | 66 (73) | 18 (24) | 25 (33) |

Task B | 63 (67) | 64 (71) | 22 (30) | 24 (32) |

Task C | 69 (73) | 63 (70) | 19 (26) | 23 (31) |

Task D | 67 (71) | 54 (60) | 21 (28) | 23 (31) |

Task E | 43 (46) | 50 (56) | 18 (24) | 20 (27) |

| ||||

Training session | Follow-up examination | |||

frequency tree (n = 94) | 2 × 2 table (n = 90) | frequency tree (n = 74) | 2 × 2 table (n = 75) | |

Task A | 84 (89) | 79 (88) | 53 (72) | 57 (76) |

Task B | 83 (88) | 78 (87) | 52 (70) | 57 (76) |

Task C | 71 (76) | 65 (72) | 45 (61) | 53 (71) |

Task D | 73 (78) | 67 (74) | 42 (57) | 49 (65) |

Task E | 54 (57) | 53 (59) | 42 (57) | 48 (64) |

| ||||

Training session | Follow-up examination | |||

frequency tree (n = 94) | 2 × 2 table (n = 90) | frequency tree (n = 74) | 2 × 2 table (n = 75) | |

Task A | 62 (66) | 59 (66) | 13 (18) | 18 (24) |

Task B | 60 (64) | 58 (64) | 17 (23) | 18 (24) |

Task C | 70 (75) | 60 (67) | 15 (20) | 15 (20) |

Task D | 69 (73) | 52 (58) | 16 (22) | 17 (23) |

Task E | 46 (49) | 44 (49) | 15 (20) | 15 (20) |

### Correct transfer

Transfer of the numerical information into the visual help in the training session could be managed in 78% (n = 365/470 frequency tree) and 76% (n = 342/450 2 × 2 table) of the tasks. In the follow-up examination in 63% (n = 234/370) and 70% (n = 264/375), respectively, the information was correctly transferred into the visual helps (Table 4).

### Correct computation

The application of the Bayesian computation in the training session was correctly used in 65% (n = 307/470 frequency tree) and in 61% (n = 273/450 2 × 2 table). In the follow-up examination 21% (n = 76/370) and 22% (n = 83/375), respectively, used correct Bayesian computation (Table 4).

### Incorrect Bayesian approaches

The commonly used incorrect Bayesian approaches*

Training session | Follow-up examination | |||||
---|---|---|---|---|---|---|

total | Frequency tree | 2 × 2 table | total | Frequency tree | 2 × 2 table | |

correct positive rate/ false positive rate | 41 | 26 (63) | 15 (37) | 16 | 11 (69) | 5 (31) |

disease yes / all positives | 14 | 7 (50) | 7 (50) | 37 | 20 (54) | 17 (46) |

correct positives / disease yes | 11 | 6 (55) | 5 (45) | 22 | 11 (50) | 11 (50) |

all positives / total | 4 | 4 (100) | 0 (0) | 14 | 6 (43) | 8 (57) |

all positives / 100 | 0 | 0 (0) | 0 (0) | 6 | 6 (100) | 0 (0) |

disease yes / correct positives | 4 | 1 (25) | 3 (75) | 1 | 1 (100) | 0 (0) |

all positives/ correct positives | 4 | 0 (0) | 4 (100) | 5 | 5 (100) | 0 (0) |

not identified | 23 | 13 (57) | 10 (43) | 29 | 14 (48) | 15 (52) |

total | 101 | 57 (56) | 44 (44) | 130 | 74 (57) | 56 (43) |

## Discussion

Differences between the 2 × 2 table and the frequency tree groups were neither meaningful nor statistically significant with regard to the primary outcome measure of correct calculation of the positive predicted values. In the training session the majority of participants were able to calculate the positive predictive value of all tasks correctly. In the reexamination after 4 weeks the proportion of participants with solutions of all tasks decreased to 26% in both groups. The transfer of the numerical information into the visual helps was comparable between the two sessions. However, participants had major difficulties in applying the correct computation as a precondition of a correct solution.

In all our tasks we have used frequency formats following the recommendation of Gigerenzer & Hoffrage [4]. In those earlier studies the frequency tree without caption has been used and we adopted this format of the frequency tree in our study. However, in more recent studies a captioned frequency tree has been used [11]. Therefore, we cannot exclude that when comparing the 2 × 2 table with a captioned frequency tree the results might be different.

Our study is the first that has compared the two visual helps 2 × 2 table and frequency tree. Previous studies have concentrated on teaching methods using either one of the visual helps or both in combination [4, 12]. These previous studies addressed different target groups, mainly medical students and physicians and focused different questions. In contrast, we addressed non-medical students without prior statistical knowledge as a first approach to lay people. Therefore, the overall results of our study are difficult to compare to previous publications.

The primary aim of our study was not to investigate different teaching methods for computing predictive values. We have tried to apply the most appropriate method according to actual research at the initiation of the study. However, overall performance of our students was poor. In the training session 58% of participants were able to calculate the positive predictive value of 4 or 5 tasks correctly. In the follow-up examination after 4 weeks the proportion of correct solutions in 4 or 5 tasks decreased to 26%. In addition, after 4 weeks participants had major difficulties in applying the correct computation as a precondition of a correct solution whereas there was only a minor deterioration with respect to the transfer of the numerical information into the visual helps.

A recent study used a computerized tutorial programme to teach Bayesian inference [11]. Within the study carried out in a rather small sample of mostly medical students, the role of the graphical aids captioned frequency tree presenting data as natural frequencies versus probability tree presenting data as probabilities in teaching Bayesian inference was explored. After 3 month participants who used the frequency tree reached 100% Bayesian solutions compared with 57% of participants using the probability tree. The authors hypothesized that it is much more important whether the proper representation is used than which graphical aid is applied [11]. Kurzenhauser & Hoffrage studied the effects of a classroom tutorial using both visual helps to teach Bayesian reasoning [12]. They achieved 47% correct answers after 2 months. Participants of the study were medical students in their second and third semester.

Generalisability of the results with respect to the overall correct solutions of our study may be limited by the prevalent innumeracy that has lately been ascertained for Germany within the OECD Programme for international student assessment (PISA). Mathematics literacy was stated to be poor in Germany especially in girls [13]. A high percentage of participants in our study were women which corresponds to the distribution of students. Transferring the self-learning tutorial to people without general qualification for university entrance would probably result in an even lower amount of correct solutions.

## Conclusions

In conclusion, our findings do not support the hypothesis that the 2 × 2 table is more valuable to facilitate the calculation of positive predictive values than the frequency tree. Regardless which visual help is used there is a need for improvement of teaching methods to approach lay people who want to participate in medical decision making.

## Declarations

## Authors’ Affiliations

## References

- Gigerenzer G: Reckoning with risk: Learning to live with uncertainty. 2002, London: Penguin BooksGoogle Scholar
- Eddy DM: Probabilistic reasoning in clinical medicine: problems and opportunities. In Judgment under uncertainty: heuristics and biases. Edited by: Kahnemann D, Slovic P, Tversky A. 1982, Cambridge: Cambridge University press, 249-267.View ArticleGoogle Scholar
- Steurer J, Fischer JE, Bachmann LM, Koller M, Ter Riet G: Communicating accuracy of tests to general practitioners: a controlled study. BMJ. 2002, 324: 824-826. 10.1136/bmj.324.7341.824.View ArticleGoogle Scholar
- Gigerenzer G, Hoffrage U: How to improve Bayesian reasoning without instruction: frequency formats. Psychol Rev. 1995, 102: 684-704. 10.1037//0033-295X.102.4.684.View ArticleGoogle Scholar
- Gigerenzer G, Hoffrage U, Ebert A: AIDS counselling for low-risk clients. AIDS Care. 1998, 10: 197-211. 10.1080/09540129850124451.View ArticleGoogle Scholar
- Hoffrage U, Lindsey S, Hertwig R, Gigerenzer G: Communicating statistical information. Science. 2000, 290: 2261-2262. 10.1126/science.290.5500.2261.View ArticleGoogle Scholar
- Hoffrage U, Gigerenzer G: Using natural frequencies to improve diagnostic inferences. Acad Med. 1998, 73: 538-540.View ArticleGoogle Scholar
- Gigerenzer G: Ecological intelligence: an adaptation for frequencies. In The evolution of the mind. Edited by: Cummins DE, Allen C. 1998, New York: Oxford University Press, 9-29.Google Scholar
- Sackett DL, Straus SE, Richardson WS, Rosenberg W, Haynes RB: Evidence-based medicine: How to practice and teach EBM. 2000, Edinburgh, London, New York, Philadelphia, St Louis, Sydney, Toronto: Churchill Livingstone, 2Google Scholar
- Johnson KV: The two by two diagram: a graphical truth table. J Clin Epidemiol. 1999, 52: 1073-1082. 10.1016/S0895-4356(99)00087-6.View ArticleGoogle Scholar
- Sedlmeier P, Gigerenzer G: Teaching Bayesian reasoning in less than two hours. J Exp Psychol. 2001, 130: 380-400. 10.1037//0096-3445.130.3.380.View ArticleGoogle Scholar
- Kurzenhauser S, Hoffrage U: Teaching Bayesian reasoning: an evaluation of a classroom tutorial for medical students. Med Teach. 2002, 24: 516-521. 10.1080/0142159021000012540.View ArticleGoogle Scholar
- PISA, the OECD programme for international student assessment. [http://www.pisa.oecd.org/]
- The pre-publication history for this paper can be accessed here:http://www.biomedcentral.com/1472-6920/4/13/prepub

### Pre-publication history

## Copyright

This article is published under license to BioMed Central Ltd. This is an open-access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.