# Pooling and Comparing Noise Annoyance Scores and “High Annoyance” (HA) Responses on the 5-Point and 11-Point Scales: Principles and Practical Advice

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## Abstract

**:**

## 1. Introduction

## 2. Linear Regression Context: Conversion of Equidistant Verbal and Numerical Annoyance Scales to a Common Scale

## 3. Logistic Regression Context: Simulating an Exposure–Response Relationship for the Percentage “Highly Annoyed” (HA) according to a Specified Cutoff Point

_{below}) or above (F

_{above}) that common cutoff point within the respective category needs to be determined.

_{above}and assignment of the response 0 (HA = 0) to the fraction below this value, a logistic exposure–HA relationship can be established that accounts for the common cutoff point. The necessary steps to do so are detailed in the next section.

#### 3.1. Choice of a Common Cutoff Point and Determination of the Fractions of Responses above That Cutoff Point

_{below}) or above (F

_{above}) the cutoff point by using Equation (2).

_{below}, F

_{above,}fraction of responses above or below the cutoff point;

_{below}would be 0.64 and F

_{above}= 0.36. While Figure 3 below gives an illustration of the pertinent fractions in this example, the respective values for other scale/cutoff point combinations are listed in Table 3.

_{above}in Table 3 can also and directly be used as the value for a “weighted” HA response given the respective cutoff point. If, e.g., a respondent in a survey marked the answer “very” on the 5-point scale, instead of “extremely,” this response would only count as HA = 0.36 (for a cutoff point of 72.73%) or HA = 0.4 (for a cutoff point of 72%), respectively, instead of HA = 1. Such weighted responses can be used for merely descriptive analyses and frequency tables, such as counting the number of HA per exposure category etc., but not as a statistical weight for weighted logistic regression analysis, which seems to be a frequent misconception.

_{above}, the basic question to ask is, “How high is the fraction of ‘very’ (on the 5-point scale) annoyed respondents that score 8, 9, or 10 on the 11-point scale?” To shed some light on this, Figure 4 shows the frequency distribution of the answers on the 11-point scale for those respondents that chose “very” on the 5-point scale in a collection of independent surveys for which we obtained the response data from both the 5-point and 11-point scales. In each histogram in Figure 4, the fraction of respondents above the cutoff point of 72.73% is colored in dark green.

_{above}is, in fact, larger than 0.36. In the above (not necessarily representative) sample of studies, the average F

_{above}is about 0.5. Indeed, a robust estimate of F

_{above}is crucial for simulating a cutoff point of 72.73% with responses on the 5-point scale, as will be discussed in the next section.

#### 3.2. Determination of the Exposure–Response Relationship for %HA for an Arbitrary Cutoff Point

_{above}can be considered HA (HA = 1) (cf. Equation (1)). Consequently, the fraction expressed in the figure F

_{below}is not considered HA (HA = 0).

_{above}) of respondents that scored “very” on the 5-point scale. The procedure can be implemented in the following steps:

- From the data table containing exposure and response data from the 5-point scale, create a subtable with only those respondents that have values 0, 1, 2, or 4 on the 5-point verbal scale. Assign the binary value HA = 0 to the responses 0, 1, 2, and the value HA = 1 to response 4.
- Create a second subtable containing only the cases with value 3 (“very”) on the 5-point scale.
- Randomly sample a fraction of F
_{above}cases in that second subtable and assign these cases the binary value HA = 1, and the remaining cases a value of HA = 0. - Combine the two subtables into a new table and run the logistic regression (with formula HA ~ exposure + additional predictors, if any) using the data of this new table.
- Save resulting model coefficients and variance-covariance matrix.
- Start over at Step 3 and repeat the procedure for a certain number of iterations, e.g., 500.
- After a sufficiently large number of iterations of the above steps, the average exposure-response relationship for a cutoff of 72.73% can be simply obtained from the means of the 500 resulting model coefficient sets; in addition, confidence intervals can be calculated from the saved variance-covariance matrices.

#### 3.3. Which Value for F_{above} Is the ‘True’ One?

_{above}is the crucial parameter to mimic the exposure-response relationship given a desired cutoff point as accurate as possible. Assuming an expectation-free, i.e. uniform distribution of responses in the “very” category to fall at any value between the lower and upper bound of the continuum covered by the “very” category, F

_{above}takes the value 0.36 for a cutoff point at 72.73% (cf. Table 3). As shown above, this value is challenged by some (yet unsystematic) empirical findings and seems, on average, to be rather in the region of 0.5, at least in the surveys included in the present exercise (cf. Figure 4).

_{above}(0.34) and the empirically derived value for F

_{above}in each study whose distribution of values on the 11-point scale for the “very” annoyed is known. To draw the curves and confidence intervals, we employed the random sampling approach described above. Analyses were performed with R version 3.5.1.

_{above}(light green curve) brings the simulated curves in almost all cases closer to the “reference” curve (blue curve) than does the value of 0.36 (red curve). This observation would, of course, challenge a recommendation to generally adopt 0.36. However, from Figure 5, we also learn that the empirical value of F

_{above}can be smaller than 0.36. This makes it difficult to recommend using a particular value. Some considerations regarding that problem are discussed further below.

_{above}for other conversions than the example discussed here should refer to Table 3.

## 4. Discussion

_{above}, different exposure-response relationships for %HA were obtained. While the expectation-free (“theoretical”) value of 0.36 assumes a uniform distribution of annoyance intensity within the “very” category, we could demonstrate that the empirically obtained values can deviate considerably from the theoretical expectation. In our sample of surveys, the empirical F

_{above}values ranged from 0.28 to 0.68. It is thus generally not unproblematic to pool or compare the data from the two different scales. Our sample of surveys is probably too small and seems too heterogeneous to recommend a particular value for F

_{above}; however, there are signs that the survey average of F

_{above}is probably larger than 0.36. The potential reasons for this remain elusive and cannot be examined within the scope of this exercise. Of course, a potentially relevant factor for the variability of F

_{above}could be the language in which the annoyance questions and scale point labels are posed. In order to be able to more generally recommend a value for F

_{above}, more surveys (that have used both the 5-point and 11-point scales) would be needed. However, such an undertaking would still be quite difficult due to the paucity of available data/surveys at hand.

_{above}values also puts the seminal meta-analysis of Miedema and Vos [6] in a different light because those two authors assumed, for reasons of simplicity, that the annoyance intensity is uniformly distributed between the lower and upper bounds of a discrete category. Should F

_{above}be systematically larger than the value a uniform distribution would imply, would the EU curves [16], in fact, have underestimated %HA. However, this must remain speculative. Our preliminary recommendation is therefore, to adhere to an F

_{above}value that requires the least theoretical or empirical assumptions, i.e., the one assuming a uniform distribution. Of course, we do not discourage researchers from adopting another (higher) value based on the insights provided in the exercise presented above.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Examples of linear regression lines resulting from modeling with four different scale conversions of the 5-point scale, as listed in Table 2:

**Left,**road traffic noise in Switzerland (whole country) [7];

**center,**railway noise in Japan (Sapporo) [8];

**right,**aircraft noise in Germany (Frankfurt) [9].

**Figure 2.**Commonly used noise annoyance scales and location of the most commonly used cutoff points 60% and ≈72% (red dashed lines) that are used to define a “highly annoyed” (HA) response.

**Figure 3.**Cutoff point of 72.73% (which corresponds to the three uppermost scale points 8, 9, 10 on the 11-point scale) on the 5-point scale and assigned percentage of HA within the five categories “not at all,” “slightly,” “moderately,” “very,” “extremely.”

**Figure 4.**Distribution of responses on the 11-point scale of respondents that chose “very” (numerical value 3) on the 5-point scale in studies on road traffic, railway, and aircraft noise, with the fraction above the cutoff point of 72.73% in dark green. Data sources are from the following studies: [7,8,9,12,13,14,15].

**Figure 5.**Exposure–response curves from different noise annoyance surveys showing %HA, as derived from the 11-point scale with cutoff point at 72.73% (“reference”), as well as based on the simulation with responses from the 5-point scale (with 95% confidence intervals as shaded areas). Two simulated curves are shown, one for F

_{above}= 0.36 and one for the empirically derived F

_{above}value of the respective study, marked with an asterisk (*). Curves are based on a simple unadjusted (crude) model. Data sources are from the following studies: [7,8,9,12,13,14,15].

**Table 1.**Number of times either the 5-point, 11-point, or both annoyance scales, as well as other scale types, were used in the surveys included in the meta-analysis by Guski et al. [1].

Noise Source | 11-Point Only | 5-Point Only | Both 11-Point and 5-Point | Other Scales | Sum |
---|---|---|---|---|---|

Road traffic | 8 | 3 | 7 | 7 ^{a} | 25 |

Railway | 1 | 4 | 4 | 1 ^{b} | 10 |

Aircraft | 9 | 1 | 5 | 15 | |

Wind turbines | 0 | 0 | 1 | 1 ^{c} | 2 |

Combined sources | 1 | 1 | 3 | 5 | |

Totals | 19 | 9 | 20 | 9 | 57 |

^{a}11-point and 4-point (one study); 4-point with notice filter question (six studies);

^{b}11-point and 4-point;

^{c}4-point and notice filter question.

**Table 2.**Conversions of scale point values on 4-, 5-, 6-, 7- and 11-point scales to values on an absolute annoyance intensity scale ranging from 0 to 100, rounded to two decimals.

4-Point Numerical Scale | |||||||||||

Numeric value: | 0 | 1 | 2 | 3 | |||||||

Upscaled value: | 0 | 33 | 67 | 100 | |||||||

Lower bound: | 0.00 | 25.00 | 50.00 | 75.00 | |||||||

Midpoint: | 12.50 | 37.50 | 62.50 | 87.50 | |||||||

Upper bound: | 25.00 | 50.00 | 75.00 | 100.00 | |||||||

5-Point Verbal Scale (ICBEN Scale) | |||||||||||

Scale point label: | “Not at all” | “Slightly” | “Moderately” | “Very” | “Extremely” | ||||||

Numeric value: | 0 | 1 | 2 | 3 | 4 | ||||||

Upscaled value: | 0 | 25 | 50 | 75 | 100 | ||||||

Lower bound: | 0 | 20 | 40 | 60 | 80 | ||||||

Midpoint: | 10 | 30 | 50 | 70 | 90 | ||||||

Upper bound: | 20 | 40 | 60 | 80 | 100 | ||||||

6-Point Numerical Scale | |||||||||||

Numeric value: | 0 | 1 | 2 | 3 | 4 | 5 | |||||

Upscaled value: | 0 | 20 | 40 | 60 | 80 | 100 | |||||

Lower bound: | 0.00 | 16.67 | 33.33 | 50.00 | 66.67 | 83.33 | |||||

Midpoint: | 8.33 | 25.00 | 41.67 | 58.33 | 75.00 | 91.67 | |||||

Upper bound: | 16.67 | 33.33 | 50.00 | 66.67 | 83.33 | 100.00 | |||||

7-Point Numerical Scale | |||||||||||

Numerical value: | 0 | 1 | 2 | 3 | 4 | 5 | 6 | ||||

Upscaled value: | 0 | 17 | 33 | 50 | 67 | 83 | 100 | ||||

Lower bound: | 0.00 | 14.29 | 28.57 | 42.86 | 57.14 | 71.43 | 85.71 | ||||

Midpoint: | 7.14 | 21.43 | 35.71 | 50.00 | 64.29 | 78.57 | 92.86 | ||||

Upper bound: | 14.29 | 28.57 | 42.86 | 57.14 | 71.43 | 85.71 | 100.00 | ||||

11-Point Numerical Scale (ICBEN Scale) | |||||||||||

Scale point label: | “Not at all” | “Extr.” | |||||||||

Numeric value: | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Upscaled value: | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |

Lower bound: | 0.00 | 9.09 | 18.18 | 27.27 | 36.36 | 45.45 | 54.55 | 63.64 | 72.73 | 81.82 | 90.91 |

Midpoint: | 4.55 | 13.64 | 22.73 | 31.82 | 40.90 | 50.00 | 59.09 | 68.18 | 77.27 | 86.36 | 95.50 |

Upper bound: | 9.09 | 18.18 | 27.27 | 36.36 | 45.45 | 54.55 | 63.64 | 72.73 | 81.82 | 90.91 | 100.00 |

**Table 3.**F

_{below}and F

_{above}for different scales and cutoff points for a uniform and expectation-free distribution of the annoyance score (annoyance intensity) value within the respective category.

Scale | Desired Cutoff Point | Cutoff Point is in Category | F_{below} | F_{above} |
---|---|---|---|---|

5-point | 60% | “very”/3 | 0.00 | 1.00 |

5-point | 72% | “very”/3 | 0.60 | 0.40 |

5-point | 72.73% | “very”/3 | 0.64 | 0.36 |

11-point | 60% | “6”/6 | 0.60 | 0.40 |

11-point | 72% | “7”/7 | 0.92 | 0.08 |

11-point | 72.73% | “8”/8 | 0.00 | 1.00 |

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**MDPI and ACS Style**

Brink, M.; Giorgis-Allemand, L.; Schreckenberg, D.; Evrard, A.-S.
Pooling and Comparing Noise Annoyance Scores and “High Annoyance” (HA) Responses on the 5-Point and 11-Point Scales: Principles and Practical Advice. *Int. J. Environ. Res. Public Health* **2021**, *18*, 7339.
https://doi.org/10.3390/ijerph18147339

**AMA Style**

Brink M, Giorgis-Allemand L, Schreckenberg D, Evrard A-S.
Pooling and Comparing Noise Annoyance Scores and “High Annoyance” (HA) Responses on the 5-Point and 11-Point Scales: Principles and Practical Advice. *International Journal of Environmental Research and Public Health*. 2021; 18(14):7339.
https://doi.org/10.3390/ijerph18147339

**Chicago/Turabian Style**

Brink, Mark, Lise Giorgis-Allemand, Dirk Schreckenberg, and Anne-Sophie Evrard.
2021. "Pooling and Comparing Noise Annoyance Scores and “High Annoyance” (HA) Responses on the 5-Point and 11-Point Scales: Principles and Practical Advice" *International Journal of Environmental Research and Public Health* 18, no. 14: 7339.
https://doi.org/10.3390/ijerph18147339