# Load-Exertion Tables And Their Use For Planning – Part 5

### Previous Part:

Load-Exertion Tables And Their Use For Planning – Part 4

## Introduction and Summary Thus Far

In the previous parts of this article series, we have covered a lot of ground. Let’s summarize it before we move forward. In short, the phenomena we want to map or model are the following: the more weight on the bar, the less reps one is able to do. To make this relationship more *generalizable* (i.e., to more individuals), weight or load on the bar is often *normalized* using the %1RM. This is what I have called *reps-max relationship* (relationship between max reps and %1RM). Even as such, this represents individual- and exercise-specific relationships that can be modeled with various mathematical equations. In the previous part, we have explored (1) Epley’s, (2) Modified Epley’s, (3) Linear or Brzycki’s, as well as (4) simple linear regression and polynomial regression equations.

Depending on our objectives (i.e., how we intend to use the estimated relationship), we can apply the aforementioned equations using various approaches. Ideally, we want the reps-max relationship mapped out to be individual and exercises specific. This implies making a model for each individual and each exercise separately. This way we can get individual profiles (for a single exercise, or movement pattern) that are very useful for individualized prescription, particularly for the *strength specialists* during the upcoming strength training phases.

More than creating individualized profiles, we are often interested in making generalizable claims and mapping general or generic relationships. For example, something we can use for training prescription for a single or group of athletes across various exercises (while also being aware of the model errors). Or to compare different exercises (i.e., are we able to do more reps to failure at 80% in the upper body pulling movements versus the pushing movements?), or group of athletes (i.e., do novice, weaker, or female athletes do more reps at 80% compared to more experiences, stronger, or male athletes?). For these reasons, we can pool together multiple individuals doing single or multiple exercises. This can then be mapped out using pooled or mixed-effect/hierarchical models.

Once we have this relationship represented or mapped out, we can use it to make strength training prescriptions and progressions. Even if mathematically clear, these progression methods rely on some strong assumptions, which are often wrong but can be useful if we are flexible enough to understand them as *tools, not truths*. In the previous parts of this article series, I have explained multiple such methods of progressions: (1) deducted intensity (DI), (2) relative intensity (RI), (3) reps in reserve (RIR), and (4) % of maximum repetitions (%MR).

Hopefully, it is clear by now that we do not have THE ONE way of mapping this out and creating THE PROGRESSION we need to follow. There may be more than one correct framework that we can use. But please do not succumb to *relativism* – not all approaches are equal or equally useful. Rather, accept the *pluralism* of models – it is possible to make a rational judgment between various frameworks and to decide some be better than others. Personally, I am *pragmatist realist* and approach these progression frameworks as heuristics. Remember tools, not truth.

Regardless of the Church you pray to, the following requisites are needed and are common across all frameworks:

- You do need to know athletes’ 1RMs
- You do need to do multiple sets to failure
- You do need to have
*designated*testing sessions or phases

In this article part, I will introduce novel techniques that do not need the above pre-requisites and as such can be implemented as *embedded testing*. Embedded testing allows *reconciliation* of the *testing~training*, or *explore~exploit* complementary aspects ^{2} ^{8} ^{18}. In a non-nerd language, this allows us to both estimate 1RMs and individual/exercise profiles through collecting training (i.e., observational) data and without *disrupting* the training process. You can still implement designated testing sessions to check what can athlete manifest, but embedded testing can be used as ongoing or continuous monitoring and *Bayesian updating*, which can help us provide more individualized prescription and monitoring. More importantly, this can be implemented for both the strength-specialists and strength-generalists.

## Taking Sets to Failure, but Not Knowing 1RM

In the previous parts of this article series, we have used reps-to-failure (RTF) data for 12 athletes. But now let’s assume that we do not know their 1RMs, so we cannot calculate %1RM. We only know the weight used and the maximal number of reps (nRM) performed (Table 1).

Athlete | Weight (kg) | nRM |
---|---|---|

Athlete A | 90.0 | 6 |

80.0 | 13 | |

70.0 | 22 | |

Athlete B | 85.0 | 3 |

75.0 | 8 | |

67.5 | 12 | |

Athlete C | 107.5 | 1 |

95.0 | 4 | |

85.0 | 6 | |

Athlete D | 95.0 | 4 |

85.0 | 9 | |

72.5 | 16 | |

Athlete E | 100.0 | 3 |

87.5 | 7 | |

77.5 | 10 | |

Athlete F | 80.0 | 5 |

72.5 | 10 | |

62.5 | 18 | |

Athlete G | 92.5 | 3 |

82.5 | 6 | |

72.5 | 12 | |

Athlete H | 117.5 | 4 |

105.0 | 8 | |

90.0 | 15 | |

Athlete I | 97.5 | 5 |

85.0 | 12 | |

75.0 | 20 | |

Athlete J | 82.5 | 2 |

75.0 | 5 | |

65.0 | 9 | |

Athlete K | 92.5 | 2 |

82.5 | 4 | |

72.5 | 7 | |

Athlete L | 125.0 | 3 |

112.5 | 6 | |

97.5 | 11 |

Table 1: Reps to failure tests of the 12 athletes

Let’s visualize Table 1. Please take a look at Figure 1. Without knowing athletes’ 1RMs, we cannot *normalize* the weight and use %1RM on the x-axis. What’s worse than having this figure that looks like a bowl of spaghetti thrown against the wall, is that we can estimate neither *generic* (or group/pooled), nor individual reps-max profiles. What can be done?

Figure 1: Reps to failure tests of the 12 athletes

All the model definitions explained in the previous part of this article series utilize %1RM as a predictor variable, but as alluded a few times so far, we do not know %1RMs because we do not know athletes’ 1RMs. This puzzled me for a brief period of time. If we do not know 1RMs, maybe we can *estimate* them? What I have done, is I have introduced another model parameter: `1RM`

(ideally we should call it `est1RM`

to differentiate it from the observed 1RM). This additional parameter is being estimated together with `k`

, `kmod`

, or `klin`

parameter. Table 2 contains model definitions using %1RM (or the models we have used so far), as well as model definitions that uses absolute weight instead.

Please note that Epley’s model estimates 0RM, not 1RM. I have tried different model definition, where 1RM is estimated: `nRM = (k * 1RM + 1RM - w) / (k * w)`

. Unfortunately, I had issues with parameter estimation. Maybe in the future, I will improve it, but in the meantime use Equation 1 to estimate 1RM from 0RM using Epley’s model.

\[\begin{equation}

1RM = \frac{0RM}{k + 1}

\end{equation}\]

Equations 1

Model Name | Uses %1RM (one parameter estimated) | Uses Weight (two parameters estimated) |
---|---|---|

Epley | nRM = (1 – %1RM) / (k * %1RM) | nRM = (0RM – w) / (k * w), |

Modified Epely | nRM = ((kmod – 1) * %1RM + 1) / (kmod * %1RM) | nRM = ((kmod – 1) * w + 1RM) / (kmod * w), |

Linear/Brzycki’s | nRM = (1 – %1RM) * klin + 1 | nRM = (1 – (w / 1RM)) * klin + 1 |

Table 2: Model definitions. Original models utilize %1RM and estimate one parameter (either `k`

, `kmod`

, and `klin`

). Model definitions utilizing absolute weight, on top of estimating `k`

, `kmod`

, or `klin`

parameter, estimate additional one (i.e., `1RM`

)

To demonstrate how these models work, let’s take our Athlete C as an example. Table 3 shows estimated parameters for Epley’s, Modified Epley’s and Linear/Brzycki’s models, including estimated 1RMs and model performance metrics. Figure 2 depicts model predictions. Note that Epley’s and Modified Epley’s models have the same weight predictions, but note that they have different %1RM predictions (particularly for the observed 1RM; this is again due to weird mathematical characteristics of the original Epley’s model where 100% 1RM is at 0RM).

Model | param | 1RM (kg) | MAE (Reps) | RMSE (Reps) | maxErr (Reps) | R2 |
---|---|---|---|---|---|---|

Epley | 0.0565 | 108.5 | 0.2287 | 0.2426 | -0.3431 | 0.9861 |

Modified Epley | 0.0535 | 108.5 | 0.2287 | 0.2426 | -0.3431 | 0.9861 |

Linear | 24.0328 | 107.8 | 0.0984 | 0.1045 | -0.1475 | 0.9974 |

Table 3: Estimated model parameters and model performance for the Athlete C. MAE = mean-absolute error; RMSE = root-mean-squared-error; maxErr = maximal error; IQR = error interquartile range; R2 = variance explained

Figure 2: Model predictions for the Athlete C. Note that Epley’s and Modified Epley’s model have the same weights predictions, but note that they have different %1RM predictions (particularly for the observed 1RM; this is again due to weird mathematical characteristic of the original Epley’s model where 100% 1RM is at 0RM). Text in the boxes represent estimated %1RM using estimated `k`

, `kmod`

, or `klin`

parameter

Compared to models using %1RM, which normalizes the individuals on the x-axis scale, using weight models for creating *generic* estimates is more tricky. We have few options. The simplest option is to make individual profiles and then average the `k`

, `kmod`

, or `klin`

parameter to get the generic/pooled estimate. Better option is to use mixed-effects (or hierarchical) models (in this case non-linear mixed-effect models). Here we have two options. We can make estimated 1RM parameter to vary across individuals (i.e. to be *random* parameter). This is often called *random-intercept* model and in our case it is a wise choice (as can be seen from Figure 1). With this model, estimated `k`

, `kmod`

, or `klin`

parameter will be *fixed* (i.e., same for everyone). Second option is to make both `k`

, `kmod`

, or `klin`

and `1RM`

parameters random (this is called *random-intercept and random-slope* model). I will show you how to fit these models using the **{STM}** package ^{10} later in this article series.

### How Can This Be Used in Practice

The practical implications of these models are instantly evident. If you have …

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