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 Intest Res > Volume 14(1); 2016 > Article
Kato, Hiraoka, Nakarai, Takashima, Inokuchi, and Ichinose: Fecal immunochemical test as a biomarker for inflammatory bowel diseases: can it rival fecal calprotectin?

### Abstract

Accurate evaluation of disease activity is essential for choosing an appropriate treatment and follow-up plan for patients with inflammatory bowel disease (IBD). Endoscopy is required for accurately evaluating disease activity, but the procedures are sometimes invasive and burdensome to patients. Therefore, alternative non-invasive methods for evaluating or predicting disease activity including mucosal status are desirable. Fecal calprotectin (Fcal) is the most widely used fecal marker for IBD, and many articles have described the performance of the marker in predicting disease activity, mucosal healing (MH), treatment efficacy, and risk of relapse. Fecal immunochemical test (FIT) can quantify the concentration of hemoglobin in stool and was originally used for the screening of colorectal cancer. We recently reported that FIT is also a useful biomarker for IBD. A direct comparison between the use of Fcal and FIT showed that both methods predicted MH in ulcerative colitis equally well. However, in the case of Crohn's disease, FIT was less sensitive to lesions in the small intestine, compared to Fcal. FIT holds several advantages over Fcal in regards to user-friendliness, including a lower cost, easy and clean handling, and the ability to make rapid measurements by using an automated measurement system. However, there is insufficient data to support the application of FIT in IBD. Further studies into the use of FIT for evaluating the inflammatory status of IBD are warranted.

### INTRODUCTION

The 2 major forms of chronic IBD are CD and UC. CD is characterized by discontinuous regions of inflammation in the intestines; it occurs most frequently in the terminal ileum and colon, but can affect any part of the gastrointestinal tract from the mouth to the anus. The symptoms of CD are: abdominal pain, weight loss, and variable degrees of diarrhea. The inflammatory process of CD is transmural and, as a result, potential disease complications include intestinal fibrosis, strictures, and fistula formation. On the other hand, the inflammatory process of UC is limited to the mucosa and submucosa of the colon, with disease almost invariably involving the rectum. Diarrhea, hematochezia, tenesmus, and urgency of defecation are classic symptoms of active UC.
Both UC and CD are chronic conditions that have periods of remission and relapse. An accurate assessment of disease activity in patients with IBD is critical for appropriately managing the disease in clinical practice. In this regard, an apparent problem is that clinical indices do not always correlate with endoscopic or histological inflammation,1 and active enteric inflammation is often present in patients who do not present with any symptoms.2 Clinical indices such as CDAI for CD and Lichtiger's Clinical Activity Index for UC are mainly based on symptoms.3,4 However, these indices rely on the subjective assessment of patients, which may cause the biologic and/or endoscopic activity to be underestimated in patients with IBD.5,6 Therefore, current opinions increasingly cite the need to achieve both clinical response and endoscopic mucosal healing (MH) in the treatment of UC and CD.7,8,9,10,11,12,13,14,15
To adequately treat and manage IBD, physicians need to accurately understand the state of disease activity in each patient. Endoscopy is required to evaluate disease activity but the procedures are sometimes invasive and burdensome to patients. In addition, monitoring the mucosal status may require endoscopy to be repeated at different stages of the disease. Repeating endoscopy would add inconvenience, discomfort, and the risk of complications to the patients. Moreover, colonoscopic procedures may worsen the disease condition, even if the IBD patient is in remission.16 Therefore, alternative non-invasive methods for evaluating disease activity or predicting mucosal status are needed. In this context, blood, fecal, and radiologic assessment methods have been suggested and investigated.
As a pioneer for fecal surrogate makers, tests for fecal calprotectin (Fcal) are frequently conducted and its performance in clinical practice for IBD has been extensively investigated. In the meantime, we recently reported on the performance of a new approach to analyzing the feces of IBD patients, the fecal immunochemical test (FIT). FIT is widely used as a method to screen for colorectal cancer (CRC). Quantitative FIT can measure the concentration of hemoglobin in feces by using an antibody that targets human hemoglobin. Fcal estimates the degree of inflammation in the gut based on the amount of infiltrating inflammatory cells, whereas FIT measures the amount of blood hemorrhaging from the intestinal mucosa. Differences in the methodology that is used to evaluate mucosal status could lead to differences in the predictability of disease activity, MH, treatment efficacy, and risk of relapse in clinical cases of IBD. The aim of this review is to introduce FIT and its performance in clinical practice for IBD, and to compare the performances of FIT and Fcal.

### MECHANISM AND PROCEDURE OF FIT

Screening for CRC was previously conducted by using guaiac-based fecal occult blood tests. FIT has since replaced guaiac-based tests because FIT has higher sensitivity for the detection of colorectal neoplasia.17 Although several kinds of single-use manual kits for FIT are available, automated analyzing systems with dedicated stool-sampling kits are currently made by several manufacturers. All available systems originated from antigen-antibody linking methods, like the ELISA. Currently, there are qualitative and quantitative systems for FITs, and quantitative systems can directly measure hemoglobin concentrations by using immunoturbidimetric methods.
The sensitivity and specificity of a variety of FIT systems have been investigated in the field of CRC screening,18 but the only available data in clinical practice for IBD was collected by using the OC-sensor system (Eiken Chemical, Tokyo, Japan). The OC-sensor system is the most popular FIT system both in Japan and in Western countries for CRC screening. This system records measurements based on latex agglutination immunoturbidmetry. FIT systems (including the OC-sensor system) consist of an automated analyzer and dedicated sampling kits. The automated analyzer is available in machines set-up for several different scales, which have different performance in the amount of throughput that it can manage in a single session. Fig. 1 illustrates the complete OC-sensor DIANA, which can measure up to 150 samples in a single session.
To perform an analysis using the OC-sensor system, patients are required to collect stools by using the dedicated sampling probe (OC-hemodia sampling probe, Eiken Chemical). An 8×2 cm test-tube-shaped container holds the sampling probe (Fig. 2A), and the container is filled with 2 mL of a hemoglobin-stabilizing buffer solution that contains a latex antihuman hemoglobin antibody reagent. To collect stool, patients insert the sampling probe into several different areas of the stool sample (Fig. 2B), and then firmly place the probe back into the tube to seal it. Generally, submitted stool samples are immediately processed and examined via the automated analyzer, but the hemoglobin concentrations in the buffer are stable for at least 3 days at room temperature. After testing the container with a stool-collected sampling probe with the automated analyzer, the measurement results can be obtained within 7 minutes, without the need for any other operations by technicians.
The OC-sensor analyzer can accurately measure fecal hemoglobin concentration in 50–1,000 ng/mL buffer. Fecal specimens with a hemoglobin concentration over 1,000 ng/mL buffer can be measured after dilution. On the other hand, FIT results are inaccurate when the hemoglobin concentration is <50 ng/mL buffer, and concentrations <50 ng/mL buffer are usually categorized as a "negative result".

### 1. Performance of FIT for UC

Currently, there are few reports that describe the performance of FIT for IBD, even though several IBD physicians in Japan have used it in clinical practice. In Japan, the prevalence of FIT use in clinical practice for IBD may be attributable to the widespread availability of FIT automated analyzers among major hospitals, as they are used for the purpose of CRC screening. Most IBD physicians that conduct FITs use it to monitor disease activity in patients at each patient visit.
The seminal report regarding the performance of FIT in clinical practice for IBD was published in 2013.19 In this report, a comparison between the results of a FIT and colonoscopy in UC patients was performed, and it was shown that FIT results accurately reflected the mucosal status in UC (Fig. 3). The main purpose of the study was to show the predictability of FIT for MH, and a negative FIT result predicted MH, which was defined as Mayo endoscopic subscore (MES)=0, with 92% sensitivity and 71% specificity.

### 2. Performance of FIT for CD

The performance of FIT for CD was examined in a cohort that was smaller than the one for UC.20 CD can involve the small intestine, and it is generally difficult to evaluate the activity in the small intestine with precision. In that study, to attain the most accurate evaluation of the small bowel, CD patients with small bowel lesions were examined with balloonassisted enteroscopy. The results showed that the FIT results significantly correlated with endoscopic activity in CD (Spearman rank correlation coefficient [r]=0.54, P<0.0001). However, the correlation was poor in patients who did not have colonic involvement (r=0.42, P=0.051), perhaps because FIT is optimized to detect bleeding in the colorectum for the purpose of CRC screening. This idea is supported by the fact that FIT is generally less sensitive to neoplasia in the proximal colon than to neoplasia in the distal colon.21

### 1. Summary of the Fcal Performance in Clinical Practice for IBD

Calprotectin is a major protein found in the cytosol of inflammatory cells. Extensive investigation has been carried out to determine the correlation between calprotectin in feces and the disease status of IBD. Shoepher et al. reported that Fcal values were more strongly correlated with endoscopic activity in UC than clinical symptoms or blood markers, and showed that the Fcal with the cutoff value of >50 µg/g had the best performance in sensitivity (93%), specificity (71%), and accuracy (89%) for the detection of endoscopically active disease.22 Another study reported that a Fcal cutoff >250 µg/g gave 71% sensitivity and 100% specificity for active mucosal disease activity in UC.23
The correlation between Fcal and mucosal status in CD has also been investigated. There have been 3 moderate scale studies (including approximately 100 ileocolonoscopies) conducted, but each study identified a different cutoff value for predicting mucosal inflammation/healing.23,24,25 A recent study showed that levels <250 µg/g predicted endoscopic remission with 94% sensitivity and 62% specificity.23 Another study showed that a cutoff level of 200 µg/g predicted endoscopically active disease with 70% sensitivity and 92% specificity.24 In contrast, the cutoff of the third study was relatively low (70 µg/g) and the value predicted endoscopically active disease with 89% sensitivity and 72% specificity.25
Therefore, Fcal has performed relatively well in predicting the disease activity of IBD. However, there are problems that relate to variation in cutoff values and definitions of disease activity between different studies. In this context, to evaluate the performance of FIT against Fcal in IBD, a direct comparison using the same cohort is required.

### 2. Comparison of FIT and Fcal in Predictability of MH

A previous study compared the performance of Fcal and fecal hemoglobin concentration in predicting active endoscopic inflammation in patients with CD and UC.26 Fcal and fecal hemoglobin concentration were measured by using ELISA, and it was found that both Fcal and fecal hemoglobin concentration were equivalent in their accuracy for identifying active endoscopic inflammation. Although hemoglobin concentrations were measured with ELISA and not FIT, fecal hemoglobin appeared to be a promising alternative to Fcal.
Most of the studies regarding fecal markers for IBD evaluated the presence of mucosal inflammation but not MH. Consequently, those studies showed a positive correlation between values of fecal markers and endoscopic activity, and have demonstrated that fecal proteins could predict mucosal inflammation rather than MH. However, in IBD, particularly in UC, surrogate markers for active inflammation are not necessary in most clinical situations, because physicians may precisely understand the active disease based on patient symptoms alone.27 Therefore, in surrogate markers, predictability for MH is more important than predictability for active disease, because the disappearance of symptoms is not always an indication of MH. Nonetheless, studies that report predictability of Fcal for MH have relatively been scarce. Only recently, Theede et al. reported that Fcal with the cutoff 192 µg/g predicted MH, where MH was defined as MES 0, with 75% sensitivity and 88% specificity.28
Based on this background, we have shown the data of the direct comparison of predictability for MH in UC between the FIT and Fcal.29 In that study, 105 colonoscopies in 92 patients with UC were evaluated in conjunction with the FIT and Fcal results. Both FIT and Fcal results significantly correlated with endoscopic activity (Fig. 4A). In addition, FIT results correlated with the results of Fcal (Fig. 4B). Although the predictability for MH was similar between the 2 markers, FIT appears to be more sensitive than Fcal for predicting MH, when a stricter definition of MH is applied (MES 0 alone; FIT, 95% sensitivity; Fcal, 82% sensitivity).
Up to now, the definition of MH is still not well-established. In particular, it remains controversial whether MES 0 or 1, or MES 0 alone should be regarded as MH. Older reports were likely to define MH as MES 0 or 1,8,30,31,32 whereas more recent studies have defined MH as MES 0 alone.33,34,35 MH should be determined in correlation with prognosis including the risk of relapse and colectomy. In this regard, some studies reported that the prognosis of patients with MES 0 did not differ from that of patients with MES 1,8,31,32 whereas others showed a significant difference in prognosis between patients with MES 0 and those with MES 1.33 In a previous study, we also observed a significant difference in the risk of relapse between patients under each definition of MH.35 In this regard, high sensitivity to MH under a stricter definition of MH may influence the predictability of the 2 fecal markers for the risk of relapse.
We also evaluated the predictability of MH in CD of the two fecal markers by performing a similar study where a direct comparison was made.20 Although the sensitivity of FIT and Fcal for MH in CD was equivalent, FIT was less sensitive for MH in CD with lesions in the small intestine alone (r=0.42; P=0.051 vs. 0.78; P<0.0001). A likely reason for the lower sensitivity of FIT for lesions in the small intestine is its optimization for bleeding in the colorectum argued in the above paragraph, and improvement and optimization for IBD would make FIT more sensitive to lesions in the small intestine.

### 3. Comparison of FIT and Fcal in the Predictability of Therapeutic Efficacy

Changes in the value of fecal markers following therapy may be a good parameter to assess the effect of the drug on the underlying inflammation. A decrease in the value of fecal markers in response to therapy is objective evidence that the drug has a beneficial effect on gut inflammation even in patients who have little change in symptoms. In this regard, monitoring of fecal markers such as Fcal in patients with CD after the initiation of anti-tumor necrosis factor-α antibody therapy could predict treatment efficacy and MH.36
In contrast, there have been no published studies to show that monitoring fecal blood or hemoglobin levels by FIT predicts the efficacy of therapeutics. However, Japanese physicians using FIT in clinical practice are generally aware that FIT values shift to negative along with the efficacy of therapy. The clinical course of a patient where treatment efficacy was monitored by FIT is shown in Fig. 5.

### 4. Comparison of FIT and Fcal in Their Predictability of the Risk of Relapse

Fecal markers could potentially predict the risk of relapse because the markers may detect subtle inflammation in the gut, which does not manifest as patient symptoms. In the clinical practice for IBD, the most suitable application of biomarkers such as fecal markers is in monitoring asymptomatic patients, to allow the early detection of subclinical relapse. Although many physicians would consider this to be the best application of fecal markers, there are few clinical studies to prove the advantage of this approach. This may be due to the difficulty in constructing the most appropriate study design.
We previously reported that FIT results became higher at 1 or 2 months prior to clinical relapse in some UC patients.37 This is illustrated in Fig. 6, which shows the clinical course and FIT results of a patient where the increase in FIT value is apparent before relapse. The ability of Fcal to predict relapse was also reported.38,39 However, in those studies Fcal in each patient was not frequently measured, perhaps due to its high cost. Recently, De Vos et al. reported that consecutive Fcal measurements in patients with UC who were undertaking infliximab therapy allowed the detection of relapse at 3 months prior to the appearance of symptoms.40 To prove whether the FIT or Fcal performs better in the earlier detection of subclinical relapse, well-designed clinical studies are needed.

### 5. FIT vs. Fcal: Other Critical Issues

IBD lasts for a lifelong period, and patients often experience disease relapse at some point in the duration of the disease. Therefore, repeat monitoring of disease activity (by using any method) is necessary. Repeated examinations may physically and economically burden the patients. Here, we compare FIT and Fcal in terms of its patient-friendliness, including factors such as test stability, cost, and examination speed.
One of the advantages of the FIT over Fcal is the test stability. The cutoff values of Fcal for the presence or absence of inflammation varies among studies and the kits that are used. In contrast, the cutoff value of FIT is relatively stable, and is almost equivalent to the cutoff used in CRC screening. In addition, the predictive performance of FIT does not vary largely as a result of the cutoff value selection. Due to the test stability, FIT can be used in a similar way amongst different institutes, and the results can be held in common among physicians. Thus, the universal nature of the FIT is a great advantage.
Because Fcal is analyzed by ELISA, the cost of one sample is relatively high, at approximately $180/sample. In contrast, the cost of a FIT is only approximately$20/sample, and most of the cost is attributable to the cost of the sampling tube. Although the automated analyzer is relatively expensive (approximately \$50,000), the equipment could be provided on a lease. A lower examination cost allows more frequent assessments to be conducted, resulting in more meticulous follow-up of patients and the possible decrease in cases of severe relapse.
High speed, high throughput, and the ability to obtain results onsite are some advantages of the FIT, although the speed of Fcal examination has also increased, compared to that during its use in the past.41,42 FIT results can be obtained within 10 minutes and physicians can refer to the results in making a medical decision during the patient visit. In contrast, Fcal takes more than several hours by ELISA. In this regard, FIT is more practical than the rather pedantic method of Fcal examination.
Lastly, the automated system and dedicated sampling kits containing buffer are unique for FIT. This packaged system has several advantages. First, sampling errors are less likely to occur. The sampling kit of any FIT system usually includes detailed instruction for collecting stools, and sampling stool by using the probe from any kit is simple for anyone to do. Second, the submitted samples can be measured by a specific automated analyzer simply by setting the kit tube into the equipment. Therefore, laboratory technicians do not need to touch stools and buffers, and measurement errors are less likely to occur. These advantages of FIT would yield stable and reliable results, which is very important in clinical practice.

### PERSPECTIVES IN THE FUTURE

In the field of CRC screening worldwide, guaiac-based manual measurement of fecal occult blood has been replaced by FIT with automated measurement systems, because of their higher performance, ease of use, low-cost, and cleanliness. Looking into this phenomenon in the field of CRC screening, the use of FIT for IBD may be foreseeable in the near future.
First, FITs other than the OC-sensor system would be available. In the field of CRC screening, the pioneer and the most widely used quantitative system of FIT has been the OC-sensor system. Currently several manufacturers make different many FIT systems for their use in CRC screening. Future investigations would study the performance of these FIT systems for IBD.
In the field of CRC screening, the 2 or 3 day-method, i.e., analysis based on 2 or 3 fecal samples, reportedly increased the sensitivity for detection of colorectal neoplasia.43 In IBD, a similar study to evaluate how the number of examined fecal samples affects the test performance has not been conducted. Even when using the sampling kit, the collection of fecal samples appears to be more difficult in patients with IBD than in patients that undergo CRC screening, because stools from IBD patients have various forms and are sometimes watery. Therefore, future studies should analyze how well FIT performs when multiple samples are examined.
Current fecal markers, including Fcal, are measured by ELISA, which is generally performed manually. Most fecal markers are proteins that are found in stools; therefore, these markers could be measured by using automated analysis equipment similar to that in the FIT system. In fact, some manufacturers appear to be developing a machine which can simultaneously measure FIT, Fcal, and other fecal proteins in a single stool sample.
Currently, FIT does not perform sufficiently well in detecting lesions in the small intestine of patients with CD. The low ability to detect lesions in the small intestine is probably due to the low sensitivity of FIT to bleeding on the oral side of the intestine. Although the lower limit of measurement by the OC-sensor is 50 ng/mL buffer, the limit could become lower due to advances in the performance of analyzers. The development of new FIT systems by other manufacturers could allow for the detection of subtle bleeding. For the application of FIT in CRC screening, cutoffs that are too low may result in lower specificity for neoplasia, leading to higher falsepositive rates. However, for the use of FIT in IBD, lower cutoffs may not negatively impact clinical practice because the absolute value is not the most clinically relevant information. In clinical practice for IBD, changes in the FIT value over the course of the disease are more clinically relevant compared to an absolute value of FIT at a single patient visit. Therefore, if lower cutoffs could allow FIT to detect changes more accurately, it may be useful in clinical practice.

### CONCLUSIONS

Here, we provided an overview of the performance of FIT in clinical settings of IBD. Fcal is still the mainstay of the fecal markers for IBD, because the marker is now widely used in Western countries. However, in the field of CRC screening, the guaiac-based fecal test, which was widely used in Western countries, has been replaced by FIT. The application of FIT as a method for CRC screening was initially approved in Asian countries and is now widely used, because automated measurement systems have been mainly developed in Asian countries such as Japan. Since manufacturers of FIT systems have started marketing activities worldwide, FIT use has also become prevalent in areas outside of Asia. Unfortunately, researchers in Asian countries that are involved in cancer screening have not disseminated the usefulness of FIT to the world. Therefore, data of FIT for CRC screening have been presented by Western researchers who only became aware of the usefulness of FIT after marketing by the manufacturers of FIT systems. FIT appears to be promising as an alternative method to evaluate stools in the field of IBD, because the concept of evaluating inflammation is quite different from that of other methods, including Fcal and lactoferrin. The fundamental difference from earlier methods may yield more important insights in the clinical practice of IBD. We hope that more clinicians and researchers in the Asian and Western countries understand the usefulness of FIT in IBD, use the systems in clinical practice, and collect and disseminate the data to all over the world.

### NOTES

Financial support: None.

Conflict of interest: None.

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### Disease course of a patient with UC who relapsed. The fecal immunochemical test (FIT) value became higher approximately 3 weeks before clinical symptoms relapsed.

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An object with a mass of  15 kg is on a ramp at an incline of pi/8 . If the object is being pushed up the ramp with a force of  7 N, what is the minimum coefficient of static friction needed for the object to remain put?

Aug 11, 2017

${\mu}_{s} \le 0.363$

Explanation:

We are given the following information:

• $\mapsto m = 15 \text{kg}$
• $\mapsto \theta = \frac{\pi}{8}$
• $\mapsto {F}_{a} = 7 \text{N}$

Two important points about static friction...

• Recall that it is the force of maximum static friction which is proportional to the magnitude of the normal force, $n$, and that ${f}_{s} \le {f}_{s \text{max}}$

• The direction of the force of static friction acting on an object is opposite the direction that the object would move if the force of friction was not present

We can calculate the coefficient of friction at the maximum force of static friction, where our forces are in equilibrium. Keep in mind that an object remains at rest as long as ${f}_{s} < {f}_{s \text{max}}$ and that the object begins to slip when ${f}_{s} = {f}_{s \text{max}}$.

The force of friction in question then should be less than ${f}_{s \text{max}}$. Therefore, we cannot calculate the exact coefficient you are looking for; we can only give the coefficient that it must be less than (that of maximum static friction).

Conclusion: ${\vec{f}}_{s} \le {\mu}_{s} n$ (direction as necessary to prevent motion).

A diagram of the situation:

Let's define up the incline as positive.

Summing the parallel (x) and perpendicular (y) components of our forces, we have:

$\left(1\right) \text{ } \textcolor{\mathrm{da} r k b l u e}{\sum {F}_{x} = {f}_{s} + {F}_{a} - {\left({F}_{G}\right)}_{x} = \cancel{m {a}_{x}} = 0}$

$\left(2\right) \text{ } \textcolor{\mathrm{da} r k b l u e}{\sum {F}_{y} = n - {\left({F}_{G}\right)}_{y} = \cancel{m {a}_{y}} = 0}$

We also know that:

(3)" "color(darkblue)(f_(s"max")=mu_sn)

$\left(4\right) \text{ } \textcolor{\mathrm{da} r k b l u e}{{F}_{G} = m g}$

Substituting equations $\left(2\right)$ and $\left(3\right)$ into equation $\left(1\right)$, we have:

$\left(5\right) \text{ } \textcolor{\mathrm{da} r k b l u e}{{\mu}_{s} {\left({F}_{G}\right)}_{y} + {F}_{a} - {\left({F}_{G}\right)}_{x} = 0}$

We can rearrange equation $\left(5\right)$ to solve for the coefficient of static friction.

$\implies \left(6\right) \text{ } \textcolor{\mathrm{da} r k b l u e}{{\mu}_{s} = \frac{{\left({F}_{G}\right)}_{x} - {F}_{a}}{{F}_{G}} _ y}$

We can find the components of the force of gravity using trigonometry.

$\sin \left(\theta\right) = {\left({F}_{G}\right)}_{x} / {F}_{G}$

$\implies {\left({F}_{G}\right)}_{x} = {F}_{G} \sin \left(\theta\right)$

Similarly, ${\left({F}_{G}\right)}_{y} = {F}_{G} \cos \left(\theta\right)$

By equation $\left(4\right)$, we have:

• ${\left({F}_{G}\right)}_{x} = m g \sin \left(\theta\right)$

• ${\left({F}_{G}\right)}_{y} = m g \cos \left(\theta\right)$

We can substitute these equations into equation $\left(6\right)$:

color(crimson)(mu_(s"max")=(mgsin(theta)-F_a)/(mgcos(theta))

Now we can substitute in our known values:

=>mu_(s"max")=((15"kg")(9.81"m"//"s"^2)sin(pi/8)-7"N")/((15"kg")(9.81"m"//"s"^2)cos(pi/8))

$= 0.363$

$\therefore$ The coefficient of static friction is $\le 0.363$.

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# Phi function help

Printable View

• July 27th 2008, 08:40 PM
Pn0yS0ld13r
Phi function help
The Euler Phi function http://www.texify.com/img/%5CLARGE%5...phi%28n%29.gif counts the number of positive integers less than or equal to n, which are relatively prime to n.

Prove that if n is odd, then http://www.texify.com/img/%5CLARGE%5...phi%28n%29.gif, and if n is even, then http://www.texify.com/img/%5CLARGE%5...phi%28n%29.gif. I don't really know how to begin.

Here's my attempt to prove the first part, if n is odd, http://www.texify.com/img/%5CLARGE%5...phi%28n%29.gif.

Let the prime decomposition of n be http://www.texify.com/img/%5CLARGE%5...Ba_k%7D%7D.gif.

So,

http://www.texify.com/img/%5CLARGE%5...%28n%29%7D.gif

But it seems like this would work for all n, not just odd n, so my above proof must be wrong. (Headbang)
• July 27th 2008, 10:25 PM
ThePerfectHacker
Who tells thou that among the prime decomposition the p_i's are different from 2?
Instead decompose n>1 as n=2^m*p1^(a1)*...*pk^ak where m>=0
• July 28th 2008, 03:47 AM
PaulRS
See result 2(b) here
• July 28th 2008, 02:22 PM
Pn0yS0ld13r
Thank you guys for the reply. I think I got it now. :)
• July 28th 2008, 05:57 PM
NonCommAlg
recall that $\varphi(ab)=\varphi(a)\varphi(b),$ whenever $\gcd(a,b)=1.$ now if $n$ is odd, then $\gcd(2,n)=1.$ so $\varphi(2n)=\varphi(2)\varphi(n)=\varphi(n).$

if $n$ is even, then $n=2^km,$ for some $m, k \geq 1,$ with $\gcd(2,m)=1.$ therefore: $\varphi(2n)=\varphi(2^{k+1} m)=\varphi(2^{k+1}) \varphi(m)=$

$(2^{k+1} - 2^k)\varphi(m)=2(2^k - 2^{k-1})\varphi(m)=2\varphi(2^k)\varphi(m)=2\varphi(2^k m)=2\varphi(n). \ \ \ \square$


text:
Principal ideal domains¶

class sage.categories.principal_ideal_domains.PrincipalIdealDomains(s=None)

The category of (constructive) principal ideal domains

By constructive, we mean that a single generator can be constructively found for any ideal given by a finite set of generators. Note that this constructive definition only implies that finitely generated ideals are principal. It is not clear what we would mean by an infinitely generated ideal.

EXAMPLES:

sage: PrincipalIdealDomains()
Category of principal ideal domains
sage: PrincipalIdealDomains().super_categories()
[Category of unique factorization domains]

class ElementMethods

Bases: object

class ParentMethods

Bases: object

additional_structure()

Return None.

Indeed, the category of principal ideal domains defines no additional structure: a ring morphism between two principal ideal domains is a principal ideal domain morphism.

EXAMPLES:

sage: PrincipalIdealDomains().additional_structure()

super_categories()

EXAMPLES:

sage: PrincipalIdealDomains().super_categories()
[Category of unique factorization domains]


text:
Illinois Journal of Mathematics

Dense subsets of Banach $\ast$-algebras

Bertram Yood

Abstract

Some subsets of a Banach ${}^{\ast}$-algebra $A$ are shown to be dense. In the special case of the algebra of $L(H)$ of all bounded linear operators on a Hilbert space $H$, the set of all $T$ in $L(H)$ for which $T^{n}$ is quasi-normal for no positive integers $n$ is dense in $L(H)$.

Article information

Source
Illinois J. Math., Volume 43, Issue 2 (1999), 403-409.

Dates
First available in Project Euclid: 19 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1255985222

Digital Object Identifier
doi:10.1215/ijm/1255985222

Mathematical Reviews number (MathSciNet)
MR1703195

Zentralblatt MATH identifier
0940.46032

Subjects
Primary: 46K05: General theory of topological algebras with involution

Citation

Yood, Bertram. Dense subsets of Banach $\ast$-algebras. Illinois J. Math. 43 (1999), no. 2, 403--409. doi:10.1215/ijm/1255985222. https://projecteuclid.org/euclid.ijm/1255985222


text: Now it was Alice's turn. “If you want to play a game, I've got a good one for you. Just as you wanted, first one to score a hundred wins. You roll a pair of dice. If you roll doubles, I win $2$ points. If the two dice have a difference of one, I win $1$ point. If the difference is $2\text{,}$ then it's a tie. If the difference is $3\text{,}$ you win one point; if the difference is $4\text{,}$ you win two points; and if the difference is $5\text{,}$ you win three points.” Xing interrupted to say “In other words, if the difference is $d\text{,}$ then Dave wins $d-2$ points.” Alice continues “Right! And there are three ways Dave can win, with one of them being the biggest prize of all. Also, rolling doubles is rare, so this has to be a good game for Dave.”


text:
# right angled triangle

• September 22nd 2012, 09:14 AM
Stuck Man
right angled triangle
Can someone explain why F=mg sin theta ? It is only simple trigonometry although it is a mechanics example.
• September 22nd 2012, 09:28 AM
emakarov
Re: right angled triangle
Draw the right triangle that has mg as the hypotenuse and the projection of mg to the inclined plane as one of its legs. This projection (i.e., its absolute value) equals F. To find one of the acute angles in this triangle use the fact that two angles are equal if their sides are perpendicular. That is, if one angle consists of rays p and q, another angle consists of rays p' and q', p is perpendicular to p' and q is perpendicular to q', then the angle measures are equal.
• September 22nd 2012, 09:44 AM
Stuck Man
Re: right angled triangle
I can see the line F but that is all.
• September 22nd 2012, 10:04 AM
Stuck Man
Re: right angled triangle
You must mean the projection equals R. I did not understand this because I was looking at the wrong triangle but you have identified the correct one. I have not followed your explanation though.
• September 22nd 2012, 11:34 AM
emakarov
Re: right angled triangle
No, I mean the projection of $m\vec{g}$ on the inclined plane equals (in absolute value and is opposite in direction to) F. The projection of $m\vec{g}$ on the line perpendicular to the plane equals R.

Look, mathematics is simpler than some philosophy or art critique because mathematicians use a single meaning of each word or expression they use. They don't chase after creating a complex bouquet of emotions with the help of finely crafted phrases full of word play and figures of speech. Such phrases are complicated because there are many ways of interpreting them and because they have to be taken as a whole. In contrast, in mathematics, if you understand the meaning of each individual term in a sentence, it should be easy to understand the whole sentence.

What is my point? It is useless to say, "I don't understand your explanation." The only way not to understand it is not to know a particular term. For example, if you don't know how to take a projection of a vector on a line, then, yes, you won't understand the explanation as a whole. But you should be able to point to one or several words and say, "I don't understand what that phrase means." Or if you don't understand how one claim follows from another, you can ask that specific question. Such questions are legitimate (but be sure you mastered the definitions of this subject matter). Saying "I don't understand your text in its entirety" is not legitimate.


text:
## FANDOM

10,744 Pages

Gag or Knackered Man is a function defined as $$\mathrm{gag}(n) = A(n,n)$$, where $$A(x,y)$$ stands for the Ackermann function.[1]

### Sources Edit

1. Really Big Numbers


text:
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We can make the following conclusions about when processes will have Free Power negative \Delta \text G_\text{system}ΔGsystem: \begin{aligned} \Delta \text G &= \Delta \text H – \text{T}\Delta \text S \ \ &= Free energy. 01 \dfrac{\text{kJ}}{\text{mol-rxn}}-(Free energy \, \cancel{\text K})(0. 022\, \dfrac{\text{kJ}}{\text{mol-rxn}\cdot \cancel{\text K})} \ \ &= Free energy. 01\, \dfrac{\text{kJ}}{\text{mol-rxn}}-Free energy. Free Power\, \dfrac{\text{kJ}}{\text{mol-rxn}}\ \ &= -0. Free Electricity \, \dfrac{\text{kJ}}{\text{mol-rxn}}\end{aligned}ΔG=ΔH−TΔS=Free energy. 01mol-rxnkJ−(293K)(0. 022mol-rxn⋅K)kJ=Free energy. 01mol-rxnkJ−Free energy. 45mol-rxnkJ=−0. 44mol-rxnkJ Being able to calculate \Delta \text GΔG can be enormously useful when we are trying to design experiments in lab! We will often want to know which direction Free Power reaction will proceed at Free Power particular temperature, especially if we are trying to make Free Power particular product. Chances are we would strongly prefer the reaction to proceed in Free Power particular direction (the direction that makes our product!), but it’s hard to argue with Free Power positive \Delta \text GΔG! Our bodies are constantly active. Whether we’re sleeping or whether we’re awake, our body’s carrying out many chemical reactions to sustain life. Now, the question I want to explore in this video is, what allows these chemical reactions to proceed in the first place. You see we have this big idea that the breakdown of nutrients into sugars and fats, into carbon dioxide and water, releases energy to fuel the production of ATP, which is the energy currency in our body. Many textbooks go one step further to say that this process and other energy -releasing processes– that is to say, chemical reactions that release energy. Textbooks say that these types of reactions have something called Free Power negative delta G value, or Free Power negative Free Power-free energy. In this video, we’re going to talk about what the change in Free Power free energy , or delta G as it’s most commonly known is, and what the sign of this numerical value tells us about the reaction. Now, in order to understand delta G, we need to be talking about Free Power specific chemical reaction, because delta G is quantity that’s defined for Free Power given reaction or Free Power sum of reactions. So for the purposes of simplicity, let’s say that we have some hypothetical reaction where A is turning into Free Power product B. Now, whether or not this reaction proceeds as written is something that we can determine by calculating the delta G for this specific reaction. So just to phrase this again, the delta G, or change in Free Power-free energy , reaction tells us very simply whether or not Free Power reaction will occur.
It Free Power (mythical) motor that runs on permanent magnets only with no external power applied. How can you miss that? It’s so obvious. Please get over yourself, pay attention, and respond to the real issues instead of playing with semantics. @Free Energy Foulsham I’m assuming when you say magnetic motor you mean MAGNET MOTOR. That’s like saying democratic when you mean democrat.. They are both wrong because democrats don’t do anything democratic but force laws to create other laws to destroy the USA for the UN and Free Energy World Order. There are thousands of magnetic motors. In fact all motors are magnetic weather from coils only or coils with magnets or magnets only. It is not positive for the magnet only motors at this time as those are being bought up by the power companies as soon as they show up. We use Free Power HZ in the USA but 50HZ in Europe is more efficient. Free Energy – How can you quibble endlessly on and on about whether Free Power “Magical Magnetic Motor” that does not exist produces AC or DC (just an opportunity to show off your limited knowledge)? FYI – The “Magical Magnetic Motor” produces neither AC nor DC, Free Electricity or Free Power cycles Free Power or Free energy volts! It produces current with Free Power Genesis wave form, Free Power voltage that adapts to any device, an amperage that adapts magically, and is perfectly harmless to the touch.
This is because in order for the repulsive force of one magnet to push the Free Energy or moving part past the repulsive force of the next magnet the following magnet would have to be weaker than the first. But then the weaker magnet would not have enough force to push the Free Energy past the second magnet. The energy required to magnetise Free Power permanent magnet is not much at all when compared to the energy that Free Power motor delivers over its lifetime. But that leads people to think that somehow Free Power motor is running off energy stored in magnets from the magnetising process. Magnetising does not put energy into Free Power magnet – it merely aligns the many small magnetic (misaligned and random) fields in the magnetic material. Dear friends, I’m very new to the free energy paradigm & debate. Have just started following it. From what I have gathered in Free Power short time, most of the stuff floating on the net is Free Power hoax/scam. Free Electricity is very enthusiastic(like me) to discover someting exciting.

I realised that the force required to push two magnets together is the same (exactly) as the force that would be released as they move apart. Therefore there is no net gain. I’ll discuss shielding later. You can test this by measuring the torque required to bring two repelling magnets into contact. The torque you measure is what will be released when they do repel. The same applies for attracting magnets. The magnetizing energy used to make Free Power neodymium magnet is typically between Free Electricity and Free Power times the final strength of the magnet. Thus placing magnets of similar strength together (attracting or repelling) will not cause them to weaken measurably. Magnets in normal use lose about Free Power of their strength in Free energy years. Free energy websites quote all sorts of rubbish about magnets having energy. They don’t. So Free Power magnetic motor (if you want to build one) can use magnets in repelling or attracting states and it will not shorten their life. Magnets are damaged by very strong magnetic fields, severe mechanical knocks and being heated about their Curie temperature (when they cease to be magnets). Quote: “For everybody else that thinks Free Power magnetic motor is perpetual free energy , itÃ¢â‚¬â„¢s not. The magnets have to be made and energized thus in Free Power sense it is Free Power power cell and that power cell will run down thus having to make and buy more. Not free energy. ” This is one of the great magnet misconceptions. Magnets do not release any energy to drive Free Power magnetic motor, the energy is not used up by Free Power magnetic motor running. Thinks about how long it takes to magnetise Free Power magnet. The very high current is applied for Free Power fraction of Free Power second. Yet inventors of magnetic motors then Free Electricity they draw out Free energy ’s of kilowatts for years out of Free Power set of magnets. The energy input to output figures are different by millions! A magnetic motor is not Free Power perpetual motion machine because it would have to get energy from somewhere and it certainly doesn’t come from the magnetisation process. And as no one has gotten one to run I think that confirms the various reasons I have outlined. Shielding. All shield does is reduce and redirect the filed. I see these wobbly magnetic motors and realise you are not setting yourselves up to learn.
Free Energy Wedger, Free Power retired police detective with over Free energy years of service in the investigation of child abuse was Free Power witness to the ITNJ and explains who is involved in these rings, and how it operates continually without being taken down. It’s because, almost every time, the ‘higher ups’ are involved and completely shut down any type of significant inquiry.

In most cases of interest there are internal degrees of freedom and processes, such as chemical reactions and phase transitions, which create entropy. Even for homogeneous “bulk” materials, the free energy functions depend on the (often suppressed) composition, as do all proper thermodynamic potentials (extensive functions), including the internal energy.


text:
Chicago Fed Letter, No. 477, March 2023 Crossref
Unit Labor Costs and Inflation in the Non-Housing Service Sector

Inflation remains high, and it is critical to understand the components of inflation. Inflation in core goods has declined over much of the past year. Inflation in housing services remains high, but the growth in rental rates of units available for rent has fallen in recent months. This suggests measured inflation in housing services, which includes rents on both occupied and vacant units, is likely to moderate over the coming year as rents on occupied units catch up to those of vacant units.1 That leaves core services excluding housing services, which accounts for more than half of core consumption expenditures. Inflation in this sector has remained elevated over the past year. Understanding why core inflation remains elevated and when it is likely to recede thus requires understanding what is happening in core non-housing services.2

Much analysis suggests that a tight labor market is contributing to inflationary pressures: Aggregate wage growth has been high; labor force participation for some groups of workers, especially older workers, is below pre-pandemic levels; and job vacancies are high relative to the size of the labor force. The sector in which concern about continued high inflation is greatest—services excluding housing—is relatively labor intensive, suggesting a tight labor market may be contributing to inflationary pressures in that sector.

One explanation for our finding that unit labor costs may not be especially helpful for forecasting inflation is that the wages firms pay their workers adjust more slowly than the prices firms charge.

This Chicago Fed Letter focuses specifically on wage growth and its potential for identifying inflationary pressures. We make two arguments, both of which are well known among economists but which we apply to services excluding housing. First, economic theory suggests that inflationary pressures depend on how nominal wage growth compares with productivity growth. We show that growth in unit labor costs in non-housing services—wage growth minus labor productivity growth—has indeed been higher in the past year than before the pandemic. This supports the idea that service providers may be facing pressure to raise their prices, given how their costs have risen.

Second, we revisit previous work on whether growth in unit labor costs helps to forecast future inflation. Several papers have now demonstrated that, at least in the aggregate, unit labor costs are not very helpful for forecasting inflation once we account for lagged inflation—certainly not as good as inflation is for forecasting growth in unit labor costs. We largely confirm these findings using up-to-date data for both the aggregate economy and non-housing services.

These results do not mean that labor costs are irrelevant for pricing decisions. Labor costs are clearly important for pricing, especially in labor-intensive sectors. One explanation for our finding that unit labor costs may not be especially helpful for forecasting inflation is that the wages firms pay their workers adjust more slowly than the prices firms charge for their goods and services. This tends to make unit labor costs a lagging indicator of inflation. Even when inflation depends on expectations of future unit labor costs, so that labor market tightness remains a source of inflationary pressures, lagged unit labor costs may not be a useful bellwether for future inflation given how price inflation may be reacting to future cost increases. Waiting until the growth in unit labor costs has subsided to determine that inflationary pressures have abated runs the risk of missing when inflationary pressures actually eased.

## Nominal wage growth and changes in unit labor costs

As has been well documented, worker compensation has increased rapidly in the past year. One illustration of this is figure 1, which extends data reported in Powell (2022). The 12-month change in average hourly earnings of private sector workers, the 12-month change in the Federal Reserve Bank of Atlanta’s wage tracker, and the four-quarter change in the employment cost index for workers in the private sector are all higher now than prior to the pandemic.

### 1. Nominal wage growth, aggregate measures

Growth in nominal wages need not translate into higher production costs if labor productivity increases at the same rate as wages. Employers would have to pay their workers higher wages, but their workers would also produce more goods and services. A better measure of whether firms face higher costs is unit labor cost, or the nominal wage wt at date t divided by real labor productivity, meaning the ratio of real output Yt to hours worked Lt. The question is whether unit labor costs have also been growing more rapidly in the past year than in the pre-pandemic period, specifically in non-housing services.

Let $\unicode{x03C0} _{t}^{ulc}$ denote the growth in log unit labor costs, i.e., $\unicode{x03C0} _{t}^{ulc}=\ln ({{w}_{t}}/{{w}_{t-1}})-\text{ln}({{Y}_{t}}/{{Y}_{t-1}})+\text{ln}({{L}_{t}}/{{L}_{t-1}}).$

We use data on value-added output for all private service-producing industries, from which we subtract value added for housing services. These value-added measures record the dollar value of output produced in a given sector for either intermediate use in other sectors or for final consumption, minus the amount paid for inputs produced in other sectors. We then compute the growth in real value-added output of non-housing services by using the value-added deflator for all services minus the deflator for value added in housing services times the share of housing value added in the value added of all services. We also assemble data on hours and wages in private service-producing industries. For hours, we use the hours of production and nonsupervisory employees on nonfarm payrolls in private service-providing industries, which is available for a longer period than for all workers. Since there are no data on hours specifically in housing services, we cannot net these from total hours. However, the share of these in total hours worked in the service sector is likely small (and is zero for housing services from owner-occupied housing). Finally, we use two measures of wages, average hourly earnings of production and nonsupervisory workers and the total compensation measure from the employment cost index, both for workers in private service-providing industries.

Figure 2 plots the implied $\unicode{x03C0} _{t}^{ulc}$ for non-housing services, reported as four-quarter changes to be comparable to figure 1. Unit labor costs in the non-housing service sector grew at a faster rate over the past year than before the pandemic. That is, nominal wage growth has exceeded productivity growth by a larger margin on average over the past year than before the pandemic. For comparison, figure 2 also shows the four-quarter change in inflation for core PCE (personal consumption expenditures) services excluding housing. The cost increases we show may have created inflationary pressures as service providers felt compelled to offset them by raising prices.

## Unit labor costs and forecasting aggregate inflation

We now turn to the question of whether growth in unit labor costs is helpful in predicting future inflation. This question has been extensively analyzed. Papers by Gordon (1988), Mehra (1991), Campbell and Rissman (1994), Emery and Chang (1996), Hess and Schweitzer (2000), Hu and Toussaint-Comeau (2010), Bidder (2015), and Peneva and Rudd (2017) find that there is at best weak evidence that aggregate unit labor costs help to predict aggregate price inflation.3 Others, including Knotek and Zaman (2014) and more recently O’Trakoun and Merone (2022), look at whether nominal wage inflation as opposed to growth in unit labor costs can predict inflation. They also find little evidence that nominal wage growth predicts inflation, although O’Trakoun and Merone find that nominal wage growth may have helped forecast inflation in services since the pandemic.

We first look at whether growth in aggregate unit labor costs helps predict aggregate inflation using updated data that includes the pandemic and then turn to data on non-housing services. The framework we use is due to Hu and Toussaint-Comeau (2010), which in turn builds on Campbell and Rissman (1994). Let $\unicode{x03C0} _{t}^{p}$ denote price inflation. Consider the following system of equations for the change in price inflation, and the change in the growth of unit labor costs $\unicode{x03C0} _{t}^{ulc}-\unicode{x03C0} _{t-1}^{ulc}:$

$\unicode{x03C0} _{t}^{p}-\unicode{x03C0} _{t-1}^{p}={{\unicode{x03B1} }^{p}}+{{\unicode{x03B2} }^{p}}\left( \unicode{x03C0} _{t-1}^{p}-\unicode{x03C0} _{t-1}^{ulc} \right)+\sum\limits_{j=1}^{4}{\unicode{x03B3} _{j}^{p}\left( \unicode{x03C0} _{t-j}^{p}-\unicode{x03C0} _{t-j-1}^{p} \right)+\sum\limits_{j=1}^{4}{\unicode{x03BB} _{j}^{p}}}\left( \unicode{x03C0} _{t-j}^{ulc}-\unicode{x03C0} _{t-j-1}^{ulc} \right)+\unicode{x03B5} _{t}^{p}$

$\unicode{x03C0} _{t}^{ulc}-\unicode{x03C0} _{t-1}^{ulc}={{\unicode{x03B1} }^{ulc}}+{{\unicode{x03B2} }^{ulc}}\left( \unicode{x03C0} _{t-1}^{p}-\unicode{x03C0} _{t-1}^{ulc} \right)+\sum{_{j=1}^{4}}\unicode{x03B3} _{j}^{ulc}\left( \unicode{x03C0} _{t-j}^{p}-\unicode{x03C0} _{t-j-1}^{p} \right)\sum{_{j=1}^{4}\unicode{x03BB} _{j}^{ulc}}\left( \unicode{x03C0} _{t-j}^{ulc}-\unicode{x03C0} _{t-j-1}^{ulc} \right)+\unicode{x03B5} _{t}^{ulc}.$

Note that this specification considers changes in inflation and the growth in unit labor costs. Some of the papers that have looked at the same question have looked at the levels of inflation and growth in unit labor costs. For a discussion of some of the reasons for using changes in these variables, see Hu and Toussaint-Comeau (2010).

Each equation contains four terms, as well as the catch-all error terms $\unicode{x03B5} _{t}^{p}$ and $\unicode{x03B5} _{t}^{ulc}.$ The first two, ${{\unicode{x03B1} }^{p}}$ and ${{\unicode{x03B1} }^{ulc}},$ are intercepts. The last two imply that the change in inflation and the change in the growth in unit labor costs may depend on lagged changes in inflation and lagged changes in the growth in unit labor costs. If changes in the growth of unit labor costs help predict changes in inflation, the coefficients $\unicode{x03BB} _{j}^{p}$ would be statistically different from zero. Likewise, if changes in inflation help predict changes in the growth in unit labor costs, the coefficients $\unicode{x03B3} _{j}^{ulc}$ should be statistically different from zero.

Finally, the second term in each equation, $\unicode{x03C0} _{t-1}^{p}-\unicode{x03C0} _{t-1}^{ulc},$ represents an error correction term. Including this term assumes, as appears to be the case in the data, that the ratio of the price to unit labor costs, which is known as the markup, tends to grow at a constant rate over time. In that case, if the growth in the markup deviated from its long-run rate, either inflation or the growth in unit labor costs (or both) would adjust to ensure that markups kept growing at their long-run rate. If inflation adjusts, we should see ${{\unicode{x03B2} }^{p}}\lt0.$ If labor costs adjust, we should see ${{\unicode{x03B2} }^{ulc}}\gt0.$

In short, unit labor costs help predict changes in inflation if either the coefficients on lagged changes in the growth of unit labor costs or the coefficient on the markup term are nonzero in the equation for the change in inflation. Inflation helps predict changes in the growth in unit labor costs if either the coefficients on lagged changes in inflation or the coefficient on the markup term are nonzero in the equation for the change in the growth of unit labor costs.

Hu and Toussaint-Comeau (2010) estimated the system above using the GDP (gross domestic product) deflator and unit labor costs for the nonfarm business sector starting in 1960:Q1 and 1984:Q1, respectively, until 2009:Q2.4 We consider three different measures of inflation—the GDP deflator, the PCE deflator, and the core PCE deflator—using data between 1984:Q1 and 2022:Q4. Figure 3 reports the p-values of tests that the respective coefficients differ from zero. A coefficient is significantly different from zero at the 5% level if the p-value is below 0.05. We cannot reject the null that the coefficients on lagged unit labor costs are all zero in the inflation equation or the null that the coefficients on lagged inflation are all zero in the unit labor cost equation. That is, $\unicode{x03BB} _{j}^{p}$ and $\unicode{x03B3} _{j}^{ulc}$ do not appear to be significantly different from zero. The coefficients on the error-correction term suggest that when growth in the markup deviates from its long-run rate, it is unit labor costs that adjust rather than inflation. That is, $\unicode{x03B2} _{j}^{p}$ does not significantly differ from zero, while $\unicode{x03B2} _{j}^{ulc}$ clearly does. To mitigate the effects of the Covid pandemic, we changed the sample period to either end in 2019:Q4 or to omit the first three quarters of the pandemic. The results for the PCE deflator and core PCE deflator were similar. For the GDP deflator, we still could not reject that $\unicode{x03B2} _{j}^{p}$ was different from zero. For the lagged changes in the growth rate of unit labor costs, the p-values were 0.02 and 0.07 when we ended the sample early and dropped the first three quarters of the pandemic, respectively. We also tried adding an unemployment gap term and relative import prices in the two equations, which did not substantially change the results.

### 3. P-values for the null that coefficients equal 0, aggregate data, 1984:Q1–2022:Q4

Sources: Authors’ calculations based on data from the U.S. Bureau of Economic Analysis and U.S. Bureau of Labor Statistics.
Price inflation Growth in ULC
${{\unicode{x03B2} }^{p}}$ $\left\{ \unicode{x03BB} _{j}^{p} \right\}_{j=1}^{4}$ ${{\unicode{x03B2} }^{ulc}}\left\{ \unicode{x03B3} _{j}^{ulc} \right\}_{j=1}^{4}$
GDP deflator 0.99 0.67 0.00 0.30
PCE deflator 0.24 0.64 0.00 0.39
Core PCE deflator 0.20 0.42 0.00 0.37

## Unit labor costs and forecasting inflation in non-housing services

Next, we turn to the non-housing service sector: Do changes in the growth of unit labor costs help predict changes in inflation for non-housing services? Results for the aggregate may not hold for individual sectors. For example, Rissman (1995) finds that unit labor costs in manufacturing and retail trade can predict inflation. However, she focuses on aggregate inflation measures rather than inflation in the same sector. Brauer (1997) finds evidence that compensation growth in services can help predict price inflation for particular services. However, he does not look at unit labor costs, nor does he use a formal statistical test. Finally, as noted above, O’Trakoun and Merone (2022) find some evidence that wage growth may have helped to predict inflation in services, although they do not look at unit labor costs.

Unfortunately, value-added measures of output are only available from 2005:Q1. To have longer time series that are comparable to those we use for aggregate data requires a proxy measure of unit labor costs before 2005. The National Income and Product Accounts (NIPA, produced by the U.S. Bureau of Economic Analysis) offer a long time series on the sale of final services produced. This measure omits services that are produced as intermediate inputs outside the service sector and includes the value of intermediates used by the service sector but produced in other sectors. It also includes government consumption expenditures on services, such as education and national defense, that are produced by the government. We subtract government consumption expenditures from this measure, as well as the value of housing services consumed, to arrive at a measure of non-housing services consumed by the private sector.5 We then convert these nominal values into real values using the PCE services deflator minus inflation in housing services times the share of housing in services consumption. In short, our measure of productivity growth before 2005:Q1 uses an output measure that omits the net flow of intermediate goods between services and the rest of the economy as well as private services purchased by the government sector. If these missing terms are not too large and tend to grow at the same rate as value added, final sales of non-housing services should serve as a reasonable proxy for the growth rate in value added when this data is not available. We confirm that the two series have grown at the same rate on average since 2005:Q1 and are highly correlated. The series on final services also appears to be less volatile than value added, suggesting that the components of value added we omit tend to be more volatile than value added as a whole.

Turning to data on prices in the service sector, we use two different inflation measures for services, the PCE services deflator minus inflation in housing services times the share of housing in services consumption and the core PCE services deflator minus inflation in housing services times the share of housing in core services. As evident in figure 2, prices of core non-housing services over the past year increased before the growth of unit labor costs in this sector, in line with the notion that unit labor costs are typically a lagging indicator.

We estimate the same system of equations as for the aggregate data. Our results for both measures of nominal wage growth are reported in figure 4. As with the aggregate data, we cannot reject the null that neither lagged unit labor costs nor a temporarily high markup help to predict high inflation going forward at conventional significance levels. A temporarily high markup still predicts higher growth in unit labor costs rather than higher inflation. When we change the sample period to mitigate the effects of data from the pandemic, the results are the same as when we use average hourly earnings as our measure of nominal wage growth. When we use the employment cost index instead, the p-values for whether the coefficients on lagged changes in the growth of unit labor costs are different from zero are 0.05 and 0.06. However, the coefficients on these lagged terms are all negative, meaning that higher growth in unit labor costs predicts slower inflation growth. We thus find no support in the sectoral data and model specifications we use for the idea that more rapid growth in unit labor costs can help predict higher inflation.

### 4. P-values for the null that coefficients equal 0, sectoral data, 1984:Q1–2022:Q3

Sources: Authors’ calculations based on data from the U.S. Bureau of Economic Analysis and U.S. Bureau of Labor Statistics.
Price inflation Growth in ULC
${{\unicode{x03B2} }^{p}}$ $\left\{ \unicode{x03BB} _{j}^{p} \right\}_{j=1}^{4}$ ${{\unicode{x03B2} }^{ulc}}\left\{ \unicode{x03B3} _{j}^{ulc} \right\}_{j=1}^{4}$
Employment cost index
PCE services deflator 0.33 0.42 0.00 0.85
Core services deflator 0.31 0.42 0.00 0.72
Average hourly earnings
PCE services deflator 0.82 0.38 0.00 0.33
Core services deflator 0.77 0.28 0.00 0.18

## Interpretation and takeaways

We conclude with a brief interpretation of our results and some key takeaways. We find that unit labor costs do not provide significant help forecasting changes in inflation, but inflation does help forecast changes in the growth of unit labor costs. Our results mirror findings for the aggregate economy from multiple studies. These results suggest that unit labor costs are a lagging indicator: Inflation helps predict growth in unit labor costs rather than the other way around. Such patterns are consistent with macroeconomic models in which wages adjust more slowly than prices. In these models, inflation is a forward-looking variable that depends on expected future marginal costs. If prices adjust more quickly than wages, inflation will rise when firms expect higher wage growth in the future. Inflation would then move ahead of, and thus help forecast, the growth in unit labor costs. Since wages adjust more slowly, growth in unit labor costs would not be useful for forecasting future inflation. Sbordone (2002) was among the first to confirm the predictions of these models with aggregate data.

Although lagged growth in unit labor costs does not help to forecast changes in inflation, this does not mean that the labor market is unimportant for inflation or that lower inflation does not require growth in unit labor costs to fall eventually. Shocks in the labor market that lead firms to expect higher nominal wage growth in the future, and thus higher growth in unit labor costs, do lead to higher inflation in standard macroeconomic models. The point is that if wages are slower to adjust than prices, unit labor costs will rise after inflation rather than vice versa.6

Finally, since growth in unit labor costs does not predict inflation, one has to be careful in using unit labor costs to identify inflationary pressures. Given the lagging nature of unit labor costs, using the slowdown in the growth in unit labor costs to identify when inflationary pressures have abated may overestimate the point at which inflationary pressures truly eased.

We thank Dan Sullivan for helpful comments and suggestions and Parker Snipes for excellent research assistance.

### Notes

1 Adams et al. (2022) look at whether the change in rents for new leases have predictive power for rental inflation as measured by the Consumer Price Index (CPI), which is the same measure used in the Personal Consumer Expenditures (PCE) price index. They conclude it does, but that the index for new leases and existing leases may not be cointegrated in the way theory suggests.

2 In his November 30, 2022, speech, Federal Reserve Chair Jerome Powell highlighted the role of core services excluding housing in contributing to inflation as measured by the core PCE deflator.

3 Ghali (1999) does find evidence that unit labor costs help forecast inflation, but these findings are unusual in the literature.

4 Some have proposed using trend unit labor costs rather than the raw labor cost to eliminate noise in measured productivity. When Peneva and Rudd (2017) use the trend of unit labor costs measure we use, they find limited pass-through from wages to inflation after the 1980s. Work using nominal wage growth that is not affected by noise in the measure of productivity also finds wages are not especially helpful in forecasting inflation. We therefore choose to use the raw data.

5 Data on the value of final services comes from table 1.2.5 of the National Income and Product Accounts. Data on government consumption expenditures comes from table 1.5.5. Data on housing services comes from table 2.4.5U.

6 Empirical evidence summarized in Nakamura and Steinsson (2013) reports the median time between price changes is on the order of four to five months for any change in the posted price and seven to eight months if we exclude temporary sales. By contrast, evidence on wages in Grigsby, Hurst, and Yildirmaz (2021) suggests wage changes for both hourly and salary workers are heavily concentrated at 12-month intervals, with a non-negligible share of changes occurring at 24 months. Bils and Klenow (2004) report that the median time between price changes is longer for services than for goods, although even for services the time is often shorter than 12 months.

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Rissman, Ellen R., 1995, “Sectoral wage growth and inflation,” Economic Perspectives, Federal Reserve Bank of Chicago, Vol. 19, No. 4, pp. 16–28, available online.

Sbordone, Argia M., 2002, “Prices and unit labor costs: A new test of price stickiness,” Journal of Monetary Economics, Vol. 49, No. 2, March, pp. 265–292. Crossref

### Opinions expressed in this article are those of the author(s) and do not necessarily reflect the views of the Federal Reserve Bank of Chicago or the Federal Reserve System.

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