‘Expected Average Heart Rate’ (EAHR) is a formula that calculates my expected average heart rate (duh) for a run given the distance and pace. The formula can be written as follows:
Constant factor + pace in mph * pace factor + (distance adjustment) * (distance factor)
A previous Mathematical Runner article goes into more detail.
Comparing the expected heart rate to my actual heart rate for a run is a way of tracking whether I had a good run or a bad run. When the data is expressed as the ratio of Expected/Actual average heart rate, the theory is the higher the Heart Rate Ratio (HRR), the better the run.
HRR improves on the actual heart rate for tracking run quality because it eliminates, in part anyhow, some of the known factors that impact my actual heart rate. Removing the impact of pace and distance helps other, unknown factors stand out.
An indication that the formula “works” is this graph showing HRR by temperature. As expected, there is a downward trend in the HRR at both higher and lower temperatures. According to this data, my optimal temperature for a run is about 48 degrees.
I was able to tweak the formula and add an adjustment for temperature which largely removes the temperature bias.
Two other measureable factors that would impact my heart rate, my weight and total mileage, were consistent over the time period, so there is no need to adjust for those factors for the purposes of this study.
Other currently unmeasured factors, things like elevation or humidity, are theoretically knowable and could conceivably be added to further refine the Expected Heart Rate formula. Some of the variance in HRR is due to these. One day I could run a difficult trail run with extreme elevation changes, which can result in a poor ratio. The next day the ratio again reverts to a more normal range on a typical road run.
However, even if I was able to adjust for all external factors, I would not be able to measure minor internal differences that likely impact my heart rate (e.g. my level of hydration). I do notice that when I give blood, the HRR dips down by 3 points or so and recovers in a few days. This is an indication that the formula does measure internal differences that I can’t feel.
In two recent runs, over the same route at about the same pace at the same temperature and at the same weight, my actual average heart rate was 152 for one and 137 for the other. While I believe that the formula works and my body was just more efficient one day than the other, what the formula does NOT do is give me any insight as to what caused the difference. Is it just random fluctuation in the body? Or was there something going on internally on one of those days that while not apparent to me did impact the run.
Most of my HRR values are quite close to 1, which is not unexpected. This chart shows the distribution of the variance from 1 of my daily HRR values:
Since I have been working on this article, I have been paying more attention to my heart rate as I run. Typically on a run I will first check my heart rate a mile or so into a run trying to guess what it will be and am often surprised when it is higher or lower than I would have guessed given my pace and how I felt. Again, I believe this is due to internal fluctuation of things that are not apparent to me, but do impact my running.
While there are day to day fluctuations that are not explainable, taking a longer-term view does seem to reveal more. I use a 10-day rolling average to view longer-term trends since that reduces the “noise” from daily variations.
I started recording this data in 2010 at age 52 and am now age 59. Surprising, at least to me, is that the trend line increased for several years before declining. Given the expected decline in performance from age 52 to age 59 it seemed to me that the overall trend should be downward over this time period since the formula does not include an age adjustment. I have no explanation for this.
In 2012 there is an obvious downward spike. I refer to this as the “Prednisone era.” I had pinched a nerve in my back and got a prescription for Prednisone. The result was a significantly increased heart rate, both at rest and during runs. The duration was not long, but the impact was significant.
In December 2013 I ran the Honolulu Marathon. The marathon was brutal and took a lot out of me. Typically after a marathon I take one day off and resume my normal running routine and do not feel adverse effects from the marathon. After Honolulu it took over a month before I really felt as if I were running normally again. This is reflected in the heart rate ratios at the time.
In the spring of 2016 the graph hits a peak. Again, I was not doing anything differently during that time period and can’t explain the high ratios at the time. However, for whatever reason, I was running very well. I believe that my result in the 2016 Boston Marathon is evidence of running well. The day was hot and generally I do not react well to hot marathons. Many runners had dreams squashed in Boston that day. The average finishing time in 2016 was 3:55:03 compared to 3:46:28 the prior year. I ran what was for me an excellent time given the conditions. I can attribute that only to how well I was running at the time. However, as can be seen in the graph, that peak lasted only a short time before returning to a more normal range.
In recent months there has been a steep decrease in the ratio. This began in December 2017 and continued into January 2018. There was nothing specific I could point to that was causing the decrease. I was not ill. My running miles and weight were consistent with other periods. The only thing that was different was that, in general, I was much stiffer than normal. Early onset rigor mortis? Whatever was causing the stiffness has passed. In late January the ratios were on the rise and getting close to normal when I tweaked a calf. I took off nearly all of the next three weeks. After that three week period the ratio was back down, as could be expected with a three week layoff. Hopefully that ratios will get back to a more normal levels as I regain conditioning. So far there seems to be evidence that the ratios are improving.
Predictive Value
I had hoped that the heart rate ratio would be useful in predicting race performance. However, my experience indicates that’s not the case. As an example, below are the ratios for 5 consecutive years of running the Leprechaun Chase 10K. I include both the ratio for the race itself as well as the ratio just before the race. The race is always held the same time of year so training going into the race is fairly consistent.
Based on the data, one might assume that 2016 would have been a great year since going into the race, the average ratio was 1.04, indicating I was running well. However, my result that year was off from the previous three years. Also, in 2017 my running appeared to be equivalent to 2013 and 2014 yet the race result was off significantly from those years.
Sadly, this analysis as well as other similar data points led me to conclude that the heart rate ratios preceding a race are not predictive of race results. I believe this is due to the day to day fluctuations in efficiency that are not measurable nor apparent, but impact running.
Conclusions
The HRR seems to be a reasonable indicator of how I am running on any given day. However, as mentioned I get no information regarding why a given day to better or worse than average, so in that sense, the formula has not been as useful as hoped. Still, simplifying ‘run quality’ to a single, at least somewhat objective, value that can be easily logged and tracked can be useful in looking at longer term changes.
When refining the ratio, it might be useful to incorporate additional data, such as:
• Humidity
• Dew point
• Altitude
• Elevation change during the run
• Wind conditions
• Did I have a massive crap before the run? : )
I would love to hear from readers their thoughts on:
• Other factors that could be useful to incorporate into a formula like this.
• Possible insights or explanations how something similar could be used to predict the quality of upcoming races.
• Any data and formulae you track that have been useful for tracking upcoming race performance or improving your training.
While the heart rate ratio did not have the predictive value for which it was developed, I am still glad to have the data. The only running data that I am glad I don’t track is marathon expenses. I think I am happier not knowing.
